Chapter 3 Ramsey Theorems, Partitions, Combinatorial Principles

61

CHAPTER

3

RAMSEY THEOREMS, PART I T I ONS , COMB1NATORIAL P R I NC IPLES

5 1 - RAMSEY'S

THEOREM

1.1. We s t a t e f i r s t t h e i n f i n i t a r y f o r m o f t h i s theorem (1926). P a r t i t i o n t h e

k classes ( k

(unordered) p a i r s o f n a t u r a l numbers i n t o

c a l l c o l o r s . Then t h e r e e x i s t s an i n f i n i t e s e t pairs included i n

, which

set E 0

m-element s e t s , o r s e t s w i t h f i n i t e c a r d i n a l

a r e assumed t o be p a r t i t i o n e d i n t o

c o l o r s . There e x i s t s an i n f i n i t e

k

of i n t e g e r s such t h a t a l l m-element subsets o f

The s e t

which we

have t h e same c o l o r .

E

T h i s g e n e r a l i z e s t o t h e case o f m

finite)

o f i n t e g e r s such t h a t

E

E have t h e same c o l o r .

i s c a l l e d monochromatic.

E

Case o f p a i r s . P a r t i t i o n t h e non-zero i n t e g e r s

t o t h e c o l o r of t h e p a i r { O,x{

0 = 0 and l e t

0 ul,

UO

t i t i o n the integers . 1 t h e p a i r {ul,x}

0 u2,

..

x

k

into

classes according

: a t l e a s t one o f t h e s e c l a s s e s i s i n f i n i t e . L e t

1 0 u1 = u1 and p a r -

be t h e elements o f t h i s c l a s s . L e t (i2 2)

x = up

k

into

classes according t o the c o l o r o f

1 u3, 1 u2,

: a t l e a s t one o f t h e s e c l a s s e s i s i n f i n i t e . L e t

t h e elements o f t h i s c l a s s . L e t

A t t h e end we o b t a i n t h e i n f i n i t e s e t w i t h elements

1 vo =.O, v1 = ul,

( i i n t e g e r ) , s a t i s f y i n g t h e f o l l o w i n g c o n d i t i o n . For e v e r y i n t e g e r \vi,vi+l\ integer

..

, {vi,vi+?J, i

. Then

-

--

vi

have t h e same c o l o r . 0

0,1,2,

...,m-2 , and

p a r t i t i o n the integers

x

uo 0 = 0, u11 = 1,

... , Um-2 m-2

l e a s t one o f these c l a s s e s i s i n f i n i t e . L e t

ulIf, u I - ~..,

t i t i o n the integers x

element

and y m- 1

x =

be t h e elements o f t h i s c l a s s . L e t t~:-~

(i>/ m)

m-1

. into

: at = m-2

,

and p a r -

=

i n t o a f i n i t e number o f classes, by p u t -

i n t o t h e same c l a s s i f f f o r each (m-1)-element

set

, t h e s e t I augmented w i t h x and I augmented w i t h

two m-element s e t s w i t h t h e same c o l o r (which depends on f i n i t e l y many

m-1

2

,..., m-2,x)

ting

u?1

have t h e same c o l o r , which we say i s a s s o c i a t e $ t o t h e

c l a s s e s , a c c o r d i n g t o t h e c o l o r o f t h e m-element s e t { O , l

and l e t

=

, the pairs

i

Case o f m-element s e t s (m 2 2 ) . Assume t h a t t h e statement i s t r u e f o r

Put a s i d e t h e i n t e g e r s k

... , v 1.

a t l e a s t one c o l o r i s a s s o c i a t e d w i t h i n f i n i t e l y many i n t e g e r s

i : a l l p a i r s subsets o f t h e s e t o f 0

.. be

2 1 u 2 = u 2 and i t e r a t e .

I , one o f t h e s e c l a s s e s i s i n f i n i t e : l e t

I

with last y

yield

I ) . As t h e r e a r e o n l y ul-',

ul;:,

..

be t h e

THEORY OF RELATIONS

62

u;

elements o f t h i s c l a s s . L e t

and i t e r a t e t h i s w i t h t h e (m-1)-element

.

ui

s e t s w i t h l a s t element

= ui-'

vo = uo 0 = 0, v1 = u11 = 1, .. i vi = ui f o r each i n t e g e r i , s a t i s f y i n g t h e f o l -

A t t h e end we o b t a i n t h e i n f i n i t e s e t w i t h elements

~ = - m-l, ~1

v

..

and i n general

l o w i n g c o n d i t i o n . For each (in-1)-element l a s t element i s

,

vi

set

I formed o f elements

t h e m-element s e t s o b t a i n e d by a d d i n g a

v

and whose

v j ( j > i )t o

I

,

a l l have t h e same c o l o r , which we c a l l t h e c o l o r a s s o c i a t e d w i t h t h e (m-1)-element

I

set

.

By t h e i n d u c t i o n hypothesis, a p p l y t h e theorem f o r

i n f i n i t e s e t o f elements

,all

v

m - 1 : we o b t a i n an

o f whose (in-1)-element subsets have t h e same as-

s o c i a t e d c o l o r . Hence an i n f i n i t e s e t , a l l o f whose m-element subsets have t h e same c o l o r . 0 1.2. Given a r e l a t i o n

A

,a

sequence whose values a r e elements o f t h e base

s h a l l be c a l l e d a sequence i n

.

A

If

A

and d e c r e a s i n g sequences were i n t r o d u c e d i n ch.2 introduced i n ch.1

u

Let

5

5

2.4;

e x t r a c t e d sequence was

2.2.

be an a - s e q u e n c e i n a p a r t i a l o r d e r i n g

ce e x t r a c t e d f r o m

IAI

i s a p a r t i a l ordering, then increasing

u

, which

A . Then t h e r e e x i s t s an w-sequen-

i s constant, o r s t r i c t l y . i n c r e a s i n q ,

o r s t r i c t l y de-

c r e a s i n g , o r c o n s i s t i n g o f elements which a r e p a i r w i s e incomparable (mod A ) . 0 P a r t i t i o n the set o f pairs

{i,j) o f integers i n t o f o u r classes ( p u t

f i x t h i n g s f o r d i s c u s s i o n ) , by d e f i n i n g a f i r s t c l a s s by t h e e q u a l i t y a second by

I

ui

< uj

,a

ui > u . (mod A) J u j (mod A) ; t h e n a p p l y RAMSEY's theorem. 0 ui

(mod A )

t h i r d by

, and

i< j ui = u

to

j '

a f o u r t h by

1.3. FINITARY FORM OF RAMSEY'S THEOREM; MONOCHROMATIC SET m, k, p 5 m

Given t h r e e i n t e g e r s

f o r every s e t o f cardinal k

>/

t

, there

, whose

e x i s t s an i n t e g e r p+>/ p such t h a t , m-element subsets a r e p a r t i t i o n e d i n t o

p p-element subset, a l l of whose

colors, there e x i s t s a

t h e same c o l o r . I t i s c a l l e d a monochromatic Consider t h e case

m

classes i s i n f i n i t e " , a t least

l/k

=

2

.

m-element subsets have

p-element subset.

Repeat t h e p r o o f o f 1.1, b u t i n s t e a d o f "one of t h e

say "one o f t h e c l a s s e s i s l a r g e " , meaning t h a t i t c o n t a i n s

o f t h e o r i g i n a l elements. 'It s u f f i c e s t o t a k e

p+ = (kp).k(kp-l)

= p.kkp i n o r d e r t o o b t a i n , a f t e r k p - 1 o p e r a t i o n s , a sequence o f l e n g t h >/ kp o f elements v , analogous t o t h o s e i n 1.1. Thus we have a l a r g e c l a s s o f v , o f

cardinality

>/

p

.0

63

Chapter 3

1.4. RAMSEY NUMBERS p+

The l e a s t such

i n t h e p r e c e d i n g p r o p o s i t i o n i s c a l l e d a Ramsey number, denoted

(p): . T h i s l o o k s i k e t h e usual Erdos-Rado n o t a t i o n , where t h e arrow w i l l be r e placed by = o r < o r > , e t c . We g i v e s e v e r a l v a l u e s .

-

Case m = 1 have

.

I f each o f t h e

k.(p-1)

a t l e a s t one c l a s s w i t h principle" : i f

k

c l a s s e s had

k(p-1)

p

+

p (p);

k

pigeonholes, t h e n

= p

.

p = m : a p-element s e t i s necessary monochromatic, t h u s

.

(3); = 6 Consider t h e elements 1,2, (b,c)

or

+ 1 t o obtain

= k(p-1)

objects.

...,6

; p a r t i t i o n t h e edges

two c o l o r s . A t l e a s t one c o n t a i n s t h r e e edges or

(p):

1 objects are partitioned i n t o

k = 1 : a s i n g l e class, thus

Calculation o f 0

elements, t h e e n t i r e s e t would

elements. T h i s argument i s c a l l e d t h e "pigeonhole

a t l e a s t one o f t h e pigeonholes has

Case Case

p-1

elements. Hence i t s u f f i c e s t o t a k e

(l,a),

(1,2)

(l,b),

.

= m

(m):

to

(1,6)

.

(1,c)

into

Either

(a,b)

(c,a)

has t h e same c o l o r , o r t h e s e t h r e e edges have t h e o p p o s i t e 2 c o l o r : t h i s shows t h a t ( 3 ) 2 i 6 . 2 To see t h a t ( 3 ) 2 > 5 , t a k e t h e usual pentagon w i t h one c o l o r , and t h e s t a r r e d pentagon w i t h t h e o p p o s i t e c o l o r . 0 Calculation o f

0

(3);

= 17

Consider t h e elements

(GLEASON, GREENWOOD 1955). 1,2,.

..,17

and p a r t i t i o n t h e 16 edges

(1,2)

to

(1,17)

(l,al), ... , ( l , a 6 ) . I t remains t o p a r t i t i o n t h e edges (ai,a.) ( i , j = 1 t o 6) i n t o two c o l o r s : hence we J 2 2 f a l l back t o t h e case ( 3 ) 2 = 6 ; t h i s shows t h a t ( 3 ) 3 6 17 . i n t o t h r e e c o l o r s . A t l e a s t one c o n t a i n s 6 edges, say

The f o l l o w i n g counterexample shows t h a t

1+1 = 0

o f t h e i n t e g e r s 0 and 1 w i t h

2 (3)3

>

16

. Consider t h e

t h e r i n g o f p o l y n o m i a l s on t h i s f i e l d w i t h t h e i d e n t i t y (0 o r 1)

composed o f 16 elements

These elements a r e e x a c t l y element i s a power

xi

+

(0 o r 1).x

x4 = x + l

.

, and

This r i n g i s

+ ( 0 o r 1).x 2 + (0 o r l ) . x 3

.

2 ,. . .,x14 (we have x15 = 1). Every non-zero

O,l,x,x

(i= O , l ,

f i e l d composed

( t h e f i e l d o f t h e i n t e g e r s modulo 2)

..., 14) , and

has i n v e r s e

x15-i

. Hence

this ring

i s a f i e l d . P a r t i t i o n the p a i r s o f polynomials i n t o three colors, according t o whether t h e d i f f e r e n c e o f these two p o l y n o m i a l s i s a cube i s o f the form

x3'+'

or

x3'+'

.

x3" ( u = O , l ,

...,4)

or

It s u f f i c e s t o see t h a t t h e sum o f two non-zero

cubes i s n o t a cube. 0 1.5. L e t

E be a f i n i t e s e t ; p a r t i t i o n i t s m-element subsets i n t o k c o l o r s i n t e g e r s p1 ,..., pk 3 m , by (pl ,..., pk)m we denote

u1 ,..., uk . Given k the l e a s t cardinal o f

E

f o r which t h e r e e x i s t s e i t h e r a pl-element

subset

THEORY OF RELATIONS

64 with color tion

(p

t h e ramsey number Calculation o f 0

.

u l , ... , o r a pk-element subset w i t h c o l o r uk

,,...,pk)m

i s symmetric. Moreover, t a k i n g (p)!

(p,.

=

(3,412 = 9

. . ,p) m .

1

=

..*

The f u n c -

pk = p, we o b t a i n

=

. .

9

We show t h a t t h i s number i s

E i t h e r among t h e 8 edges

p

(1,2)

to

J o i n up t h e i n t e g e r s 1 t h r o u g h 9 by edges. (1,9)

t h e r e e x i s t 4 edges o f c o l o r ( + ) . T h i s

t h e n y i e l d s e i t h e r a 3-element s e t w i t h c o l o r (+) o r a 4-element s e t w i t h c o l o r ( - ) , O r t h e r e e x i s t 6 edges w i t h c o l o r ( - ) , which t h e n y i e l d s e i t h e r a 3-element subset

(+) o r a 4-element subset ( - ) . Or f i n a l l y none o f t h e p r e c e d i n g cases i s r e a l i z e d f o r any o f t h e p o i n t s

1 through

9

. Then

f r o m each p o i n t t h e r e emanate e x a c t l y

3 edges (+) and 5 edges ( - ) . B u t t h i s i s i m p o s s i b l e , s i n c e we would t h e n have 3.(9/2) = 27/2 edges ( + ) . 2

>

2 C a l c u l a t i o n o f ( 4 ) 2 = 18

.

We now show t h a t t o t h e edge

.

8 Take t h e i n t e g e r s 0 t o 7, and g i v e t h e c o l o r (+) i f f t h e a b s o l u t e v a l u e o f y - x i s 3, 4, or 5. 0

(3,4)

(x,y)

Take t h e i n t e g e r s 1 t o 18. Among t h e edges emanating f r o m 1, t h e r e a r e a t l e a s t

0

9 o f t h e same c o l o r which we d e s i g n a t e ( + ) . They j o i n 1 t o t h e i n t e g e r s d e s i g n a t e d

...,

al,. . . , ag . By t h e preceding, i n t h e s e t o f al, ag t h e r e e x i s t s e i t h e r a 3-element s e t w i t h c o l o r ( + ) , o r a 4-element s e t o f t h e o p p o s i t e c o l o r ( - ) . Hence t h i s Ramsey number i s a t most 18.

\<

The f o l l o w i n g ewample w i l l prove t h a t t h e Ramsey number i s n o t

17. Take t h e

i n t e g e r s modulo 17, so 0 t o 16. For any two d i s t i n c t x, y i n t h i s s e t , we g i v e t o t h e edge (x,y) t h e c o l o r (+) i f f x i s congruent t o y modulo a q u a d r a t i c r e s i d u e , so mod

1, +, 2,

f 4,

o r 2 8 ; t h e c o l o r ( - ) i n t h e o p p o s i t e cases.

Suppose t h a t t h e r e e x i s t 4 i n t e g e r s

a,b,c,d

same c o l o r . We can r e p l a c e these i n t e g e r s by

such t h a t a l l t h e i r edges have t h e and 0 , and t h u s

a-d, b-d, c-d

.

can c o n s i d e r o n l y t h e case o f O,a,b,c We can r e q u i r e t h a t t h e 6 i n t e g e r s a, b, c, b-a, c-b, a-c be non-zero and e i t h e r a l l r e s i d u e s o r a l l non-residues. M u l t i p l y i n g by t h e i n v e r s e o f the 5 integers

a

, we

can reduce t h i s t o t h e case o f

b, c, b-1, c-1, c-b

r e s i d u e s . Then t h e p o s s i b i l i t i e s f o r b = -1 c

-

, we

b # -7

we o b t a i n

have

c

b

and

c

# -1 , c # 2 s i n c e c-b # 3

(3,5)L = 14

-1, +2, -8 c # -8 s i n c e

a r e reduced t o

. Moreover

. The same argument f o r c = -1 : i m p o s s i b l e . F o r b-c = -7 : i m p o s s i b l e . 0

Calculation o f

0, 1, b, c

with

which a r e a l l non-zero and a l l q u a d r a t i c

b = 2

and

. For

c = -8

.

Take a s e t w i t h 14 p o i n t s and l e t a be i n t h i s s e t . E i t h e r f r o m a t h e r e eman a t e a t l e a s t 5 edges w i t h c o l o r (+), which y i . e l d s a 3-element s e t monochromatic

0

Chapter 3

65

w i t h c o l o r ( + ) , o r a 5-element monochromatic s e t w i t h c o l o r ( - ) . O r f r o m

a

t h e r e emanate a t most 4 edges (+), hence a t l e a s t 9 edges ( - ) . Then s i n c e (3,4) 2 = 9 , t h i s y i e l d s e i t h e r a (+)-monochromatic 3-element s e t , o r a (-)-monochromat i c 5-element s e t . Thus

(3,5)2

i s bounded above by 14.

To see t h a t i t equals 14, t a k e t h e 13 i n t e g e r s 0 t h r o u g h 12. Give t h e c o l o r (t) t o the p a i r

{x,y)

(where

the absolute value o f

x

and y

a r e d i s t i n c t elements among O , l ,

..., 12)

iff

equals 2, 3, 10 o r 11; c o l o r ( - ) i n o t h e r cases. 0

y-x

Other known ( o r almost known) v a l u e s o f b i n a r y Ramsey numbers: (3,6)2 = 18 : KALBFLEISCH 1964; i n d e p e n d e n t l y KERY 1964; n

(3,7)L = 23 : GRAVER, YACKEL 1968; (3,8)

2

= 28 o r 29 ; a l s o

(3,9)

2

= 36 : GRINSTEAD, ROBERTS 1982.

1.6. Below we l i s t some i n e q u a l i t i e s f o r t h e s m a l l e s t Ramsey numbers whose e x a c t value i s n o t known. B i n a r y numbers.

,<

25

(4,5)[

4

28 : l o w e r bound o f 25 by KALBFLEISCH 1964; upper

bound o f 28 by WALKER 1971. 42

6 (5); ,< 55

: l o w e r bound o f 42 by I R V I N G 1973; a l s o GARCIA 1975; a l s o HANSON

1976; upper bound o f 55 by WALKER 1971. 2 128 4 ( 4 ) 3 d 254 : l o w e r bound o f 128 by HILL, I R V I N G 1982; upper bound o f 254 e a s i l y o b t a i n e d v i a (2,4,4) 2 = (4,4) 2 = 1 8 and (2,3,4) 2 = (3,4) 2 = 9 ; t h e n (3,3,4)26 51

6

(3);

34

and

6 65

(3,4,4)2

6

85

.

: l o w e r bound o f 51 by CHUNG 1973; upper bound of 65 by FOLKMAN

1974, a l r e a d y announced by WHITEHEAD 1973. 6 ( 3 ) 52 322 : l o w e r bound 159 by FREDRIKSON 1979; t h e upper bound i s easy.

<

159

S t i l l c o n s i d e r i n g b i n a r y Ramsey numbers, we have t h e easy i n e q u a l i t y : 2 (3)k

1) + 2 , which y i e l d s

2 (3)k

4

k ! e + 1 ( c l a s s i c a l number e ) .

I n t h e o t h e r d i r e c t i o n , we e a s i l y o b t a i n

2 (3)k

3

(3

5

2 k((3)k-1

-

on Schur’s numbers. Improved by 2 (3)4

>/

Moreover ratio:

2 (3)k+13

2 3.(3)k

+

k

+ 3)/2

2 (3)k-2

-

: see e x e r c i s e 1

3

( f o r example

2 3.(3)3 = 3.17 = 51 ); see CHUNG 1973. 2 ( ~ + 1 i)s~bounded above, up t o a c o n s t a n t f a c t o r , by t h e f o l l o w i n g (2p)!(Log Log p ) / (p!)[Log p

Ternary number.

13

4

(4);

bound o f 15 by GIRAUD 1969.

6

: see YACKEL 1972.

15 : l o w e r bound o f 13 by ISBELL 1969; upper

66

THEORY OF RELATIONS

2 - LEXICOGRAPHICALLYORDERED THEOREM, NASH-W ILLI AMS ' THEOREM §

SET,

GALVIN'S

INITIAL

INTERVAL

2 . 1 . LEXICOGRAPHICALLY ORDERED SET, LEXICOGRAPHIC RANK Totally order the s e t of f i n i t e s e t s of integers lexicographically, by f i r s t difference: ( s e t a ) ,< ( s e t b ) i f f the l e a s t integer in a i s s t r i c t l y l e s s than the l e a s t integer in b ; or in the case of equality, compare the second l e a s t integer of a with the second l e a s t integer of b , e t c . The empty s e t i s defined

t o precede a l l other s e t s in t h i s ordering. Finally i f a i s a proper i n i t i a l interval o f b , we p u t a < b . of f i n i t e s e t s of integers i s said t o be lexicographically well-ordered, A set i f the lexicographic ordering of elements o f i s a well-ordering. The corresponding order type i s called the lexicographic rank of F . I t i s a countable ordinal. For example, the s e t of singletons of integers i s lexicographically well-ordered with rank w, the s e t of pairs with rank U 2. The s e t o f a l l f i n i t e s e t s of integers i s n o t lexicographically well-ordered. Indeed we obtain a lexicographically decreasing W-sequence, by taking the singleton i l } , then the p a i r { 0 , 2 ) , , etc. then {0,1,3} , then {0,1,2,4} be a s e t of f i n i t e s e t s of integers, such t h a t : Let are mutually incomparable under inclusion; (1) the elements of ( 2 ) every i n f i n i t e s e t of integers includes a subset which belongs t o 3 ; i s lexicographically well-ordered (communicated by POUZET in 1980). then t o be lexicographically well-ordeNotice t h a t ( 2 ) alone i s not s u f f i c i e n t f o r red: take the s e t of a l l the f i n i t e s e t s of integers; (1) alone i s not s u f f i c i e n t : take the above decreasing W-sequence. Consider a non-empty subset of and show t h a t there e x i s t s a minimum f o r the lexicographic ordering. Let a. be the l e a s t integer such element in t h a t there e x i s t s an element of beginning with a. I f the singleton ao) belongs t o , then i t i s the minimum of Otherwise, take the elements of which begin with a. , and l e t al be the l e a s t integer such t h a t there e x i s t s an element of beginning with ao, a l . If the pair {ao,al} belongs t o , then i t i s the minimum of Otherwise, i f t h i s procedure never terminates, then we obtain an i n f i n i t e increasing sequence a. < al < . . < ai < . . ( i inte-

0

5

5 3

5

5

9

F,

5

5.

.

.

ger). By our hypothesis ( 2 ) , there e x i s t s a f i n i t e : denote by ah the the ai and belonging t o , ah also a f i n i t e set beginning with ao, a l , to B u t t h i s contradicts our hypothesis ( I ) of

F.

4

.

...

.

s e t composed of certain of l a s t among these. There e x i s t s and belonging t o , hence incomparability. 0

5

67

Chapter 3

2 . 2 . INITIAL INTERVAL THEOREM Let p be a s e t of f i n i t e s e t s of integers, such t ha t every i n f i n i t e s e t of Then there e xists an integers includes as a subset a t l e a s t one element of in f in i t e s e t E of integers such t h at every i n f i n i t e subset of E has an elea s a n i n i t i a l interval (GALVIN 1968; the following proof, using only -.ment of the axioms of ZF, i s due t o POUZET 1980, unpublished). Note f i r s t t h a t RAMSEY's theorem eas i l y follows from the preceding statement. Indeed, p a r t i t i o n the pairs of integers into two colors (+) and ( - ) . Then either there e x i s t s an i n f i n i t e s e t of integers a l l of whose pairs belong t o (+), or every i n f i n i t e s e t includes an element of color ( - ) . Then by the above statement, there e x i st s an i n f i n i t e s e t E such t ha t every i n f i n i t e subset of E begins with a pair belonging t o ( - ) : in other words every pair belongs t o ( - ) . To simplify the proof, note t h at i t i s always possible t o assume tha t the elements of are mutually incomparable with respect t o inclusion. Indeed, starting with an a r b i t r ar y we obtain the subset T o by taking those which are minimal with respect t o inclusion. Every i n f i n i t e s e t elements of The i n i t i a l interval theorem, of integers includes a t l e a s t one element of F' when r e s t r i c t e d by the preceding condition, says t ha t there e xists an infinite set E , such t h a t every i n f i n i t e subset of E has an element of 3", hence of F , as an i n i t i a l interval. By the preceding 2 . 1 , we see t h a t i t suffices t o prove the i n i t i a l interval theorem f o r an arbitrary lexicographically well-ordered s e t (whose elements will no longer necessarily be incomparable under inclusion). Thus we are led t o prove the following statement.

F.

3,

.

2.3. Let be a lexicographically well-ordered s e t of f i n i t e subsets of integers could contain the empty s e t as an element); then: (1) e i t h e r there e x i s t s an i n f i n i t e s e t of integers which includes no element

(3'

of F-; ( 2 ) or there e x i s t s an i n f i n i t e s e t of integers, each of whose i n f i n i t e subset has an element of 3- as i n i t i a l interval (the empty s e t i s considered as an i n i t i a l interval of every s e t ) . 0 W e argue by induction on the lexicographic rank of F . Suppose f i r s t tha t the rank i s equal t o 1 , so t h at 3 i s the singleton of a f i n i t e s e t F of integers. Then the i n f i n i t e s e t of a l l integers not belonging t o F s a t i s f i e s our conclusion (1) i f F i s non-empty, our conclusion ( 2 ) i f F i s empty. Let o( be a countable ordinal. Suppose the statement i s true f o r every s e t of lexicographic rank < o(. We shall prove i t f o r every s e t of rank n( .

THEORY OF RELATIONS

68

More strongly, in order to avoid use of the axiom of choice, or even a weakened form of choice, suppose t h a t there exists a function h which, to each ordered p a i r (E,F) where E i s an infinite set of integers, If a set of f i n i t e subsets of E of lexicographic ranks < I% , associates an infinite set h ( E , F ) E satisfying one of the conclusions (1) or ( 2 ) . More precisely, either h ( E , y ) includes as a subset no element of , or every infinite subset of h ( E , begins with an initial interval which belongs t o W e will prove that there has exists an analogous function for the ordered pairs whose second term lexicographic rank d . Thus h will be progressively extended t o all countable lexicographic ranks, so t o a l l ordered pairs (E,F) S t a r t with an infinite s e t E of integers and a set of f i n i t e subsets of E with lexicographic rank O C . For each integer i of E , denote by Fi the i s the union of subset of those elements of which begin w i t h i Hence of 3;i s the sum along w of the the and the lexicographic ordering o( lexicographic orderings of the Fi . If there exist infinitely many integers i of E with empty, then the set of these i does not include as a subset any element of F . This will be, by definition, h ( E , F ) which then verifies our conclusion (1). Consider the other case, and l e t m(0) be the least integer of E after which Fi i s never empty. Denote by Mo the set of integers of E which are >,m(O) and by the set Mo with i t s minimum m(0) removed. The Ti all have lexicographic ranks t d , hence the function h i s already defined for these. and then = the set of the elements of , each of W e put $ = whose m i n i m u m m(0) has been removed: the lexicographic r a n k of this is s t i l l < e( P u t El = h(Mg, 2 Mi . Either there exist infinitely many integers i of El for which El includes no element of Ti . Then the set o f these i i s by definition h ( E , g ) , which satisfies our conclusion (1). Or i n the opposite case, l e t m(1) be the least integer of El , from which point on every Ti restricted t o i t s elements which are subsets of El i s never empty. Denote by M1 the set of integers of El which are >/ m(1) , and Mi the set M1 with i t s minimum m(1) removed. Let = res-

s

5

F)

F.

.

.

Fi

Fi

M i

Fm(o)

.

go

50 5 0)

.

5i

5;

zl Fm(l)

tricted to elements which are subsets of M1 Then = the set of the elemnts of each w i t h i t s minimum removed: the lexicographic rank of

5

.

$1

and of i s s t r i c t l y less than o( Let E2 = h(M;, 5 Mi , and iterate this procedure. Then either, a t the end of a f i n i t e number of steps, we obtain an Er ( r integer) with infinitely many integers i of E r f o r which no element of Fi i s a subset of E r . Then by t o be the set of these i which satisfies (1). definition, we p u t h ( E , $ ) Or this described process continues indefinitely: we must consider two subcases.

5 i)

Chapter 3

69

First subcase. There e x i s t i n f i n i t e l y many integers k f o r which the s e t Take the i n f i n i t e Ek+l = h(Mi , contains as a subset no element of corresponding s e t of minimums m ( k ) . Then m ( k ' ) t Z Ek+l Mk f o r a l l k , k ' > k . , hence no So the s e t of minimums m(k) contains as a subset no element of element of F m ( k ) f o r any k Finally our s e t of m ( k ) contains as a subset no element of 8 : we take i t as our definition of h ( E , 7 ), which s a t i s f i e s ( 1 ) .

9

9 k)

.

%K

.

In the second subcase, because of the definition of the function h , there e x i s t s an integer ko from which point on, every i n f i n i t e subset o f Ek+l begins by a

% .

A f o r t i o r i , i f K denotes the s e t of minimums possibly empty element of m ( k ) f o r k >/ kg , then every i n f i n i t e subset of K begins by an element of a

k , hence by an element of s a t i s f i e s our conclusion ( 2 ) . 0

F.

Thus the s e t

K

, which

we take f o r

h(E,

3

)

2.4. NASH-WILLIAMS' THEOREM (1965)

5k

Consider two d i s j o i n t s e t s of f i n i t e s e t s o f integers. Suppose t h a t no , and vice-versa. Then element of F i s an i n i t i a l interval o f an element of there e x i s t s an i n f i n i t e s e t E of integers, which con ains a s a subset no element . NASH-WILLIAMS only assumes of F , orwhich contains as a s u b T t no element of , one i s never an i n i t i a l interval t h a t , f o r two d i s t i n c t elements of

9

p l _ l _ _ _

9 5 of the other. The present stronger statement r e s u l t s from a remark by HODGES.

-

F w

Motice t h a t RAMSEY's theorem follows. Indeed, given two d i s t i n c t pairs of integers,

or in general, f o r p a fixed integer, given two d i s t i n c t p-element s e t s of integers, one i s never an i n i t i a l interval of the other. 0 Either there e x i s t s an i n f i n i t e s e t o f integers having

no subset which i s an

element of 3 . Or every i n f i n i t e s e t of integers has a sirbset which i s an element of F . I n the l a t t e r case, by GALVIN's theorem 2 . 2 , there e x i s t s an i n f i n i t e s e t E of integers, such t h a t every i n f i n i t e subset of E has an element of T as i n i t i a l i n t e r v a l . Then E has no subset which i s an element of . Indeed, i f i t , then take an i n f i n i t e subset X of E contained as a subset an element G of with i n i t i a l interval G There e x i s t s an element F of which i s an i n i t i a l interval of X : thus F i s an i n i t i a l interval of G , or G an i n i t i a l interval of F : contradiction. 0

.

5 3 - UNCOUNTABLE E R D ~ S , RADO

CASE,

9

5

PARTITION

THEOREMS:

DUSHNIK,

MILLER,

3.1. SIERPINSKI 'S COUNTEREXAMPLE (1933) There e x i s t s a partition of the pairs of reals into two colors, such t h a t every monochromatic s e t i s countable (uses axiom of choice).

,

70

THEORY OF RELATIONS

0 Take a w e l l - o r d e r i n g

with

o f t h e s e t o f r e a l s . Then t o each p a i r o f r e a l s

A

i n t h e usual o r d e r i n g , g i v e t h e c o l o r (+) i f

x c y

x < y (mod A)

x,

y

and

t h e c o l o r ( - ) i f x > y (mod A) , The p r o p o s i t i o n f o l l o w s f r o m t h e f a c t t h a t e v e r y s t r i c t l y i n c r e a s i n g ( o r s t r i c t l y decreasing) sequence o f r e a l s i s c o u n t a b l e (see ch.1

5

4.5). 0

As a p a r t i c u l a r case of 3.4 below, we mention h e r e t h a t , f o r e v e r y p a r t i t i o n o f t h e p a i r s o f elements of a s e t o f c a r d i n a l

i n t o two c o l o r s , t h e r e i s a

a 1(uses

monochromatic subset o f c a r d i n a l

axiom o f c h o i c e p l u s t h e continuum

hypothesis). We can summarize t h i s s i t u a t i o n by u s i n g t h e n o t a t i o n f o r Ramsey numbers w i t h f i n i t e o r i n f i n i t e c a r d i n a l values. Then t h e usual Ramsey theorem i s w r i t t e n

( w ); = W f o r a l l i n t e g e r s m, k 2 w i t h 3.4 y i e l d s ( w 1 ) 2 = Ld2 . 3.2.

.

The p r e c e d i n g p r o p o s i t i o n complemented

PARTITION LEMMA (DUSHNIK, MILLER 1941)

L e t A = C d N be an i n f i n i t e r e g u l a r aleph. P a r t i t i o n t h e p a i r s o f elements o f A i n t o two c o l o r s which we d e s i g n a t e by (+) and ( - ) . Then e i t h e r , f o r e v e r y subset B which i s e q u i p o t e n t w i t h A , t h e r e e x i s t s an element

a

of

B

and a s e t o f elements

equipotent w i t h

A

.

of

B

A

>, ,a

al

in

B

a.

A ,

.

equipotent w i t h

f i r s t c o n c l u s i o n . Consider t h e elements o f the ordinals. Let

which i s e q u i p o t e n t w i t h

O r t h e r e e x i s t s a subset o f - A

, a l l o f whose p a i r s have c o l o r (+)

Assume t h e r e e x i s t s a subset

0

f B

x

have c o l o r (-)

where a l l t h e p a i r s {a,x)

B

A

, which

negates t h e

o r d e r e d by t h e usual o r d e r i n g on

B ; by h y p o t h e s i s t h e r e e x i s t s an

be t h e minimum o f

1

, such t h a t f o r e v e r y x >, al t h e p a i r { aO,x has c o l o r (+). i C w* , assume t h a t f o r t h e j < i we have a s t r i c t l y

By i n d u c t i o n , g i v e n

i n c r e a s i n g sequence o f elements

<

a

j

of

B

,

such t h a t a l l p a i r s o f t h e

a.

J

have

i , t h e r e a r e l e s s than wd many x i n B such t h a t the p a i r {aj,x) has c o l o r ( - ) . Since u4 i s r e g u l a r , t h e s e t o f a l l such x f o r a l l j < i has c a r d i n a l i t y < ud . Thus t h e r e e x i s t s an ai of B , which i s s t r i c t l y above a l l t h e a , and such t h a t f o r a l l x >, ai j and a l l j c i , t h e p a i r ( a j , x ) has t h e c o l o r ( + ) . F i n a l l y we o b t a i n an

t h e c o l o r (+). For e v e r y

j

C30(-sequence o f elements, a l l o f whose.pairs have c o l o r ( + ) . 0 Notice t h a t the proposition i s f a l s e f o r every singular cardinal be t h e c o f i n a l i t y of . For e v e r y i < l e t T < W, Ai

of

A = W4

, such

s t r i c t l y subpotent w i t h

A

r~~

.

Indeed

, t a k e a subset

t h a t t h e u n i o n o f t h e Ai i s A , b u t e v e r y Ai is . F o r e v e r y p a i r c o n t a i n e d i n an Ai , g i v e t h e c o l o r

( - ) , and f o r p a i r s o f elements b e l o n g i n g t o d i s t i n c t

Ai

's, the color (+).

71

Chapter 3

3.3. PARTITION THEOREM (DUSHNIK, MILLER 1941) Let A be an a r b i t r ar y i n f i n i t e s e t ; partition the pairs of elements of two colors (+) and ( - ) . Then ei t h er there ex i s t s a denumerable subset of

A into A A which i s equipotent

which i s (-)-monochromatic, or there ex i s t s a subset of w i t h A fi (+)-monochromatic (uses axiom o f choice). DUSHNIK and MILLER mention the influence of ERDOS. A different proof of the theorem i s given by ERDOS, RADO 1956. Using Ramsey numbers notation and replacing A by an aleph w, , we have ( ~ 3 w , W ) '= L&, Replace A by an aleph which we designate by &>,;,(axiom of choice), and assume f i r s t t h a t t h i s aleph i s regular. By the preceding lemna, i f our second conclusion i s f a l s e , then there ex i s t s an a. in A for which the s e t A. of x 7 a. (mod A ) such t h at {ao,x] has color ( - ) i s equipotent with A . Take t h i s a. minimum (mod A) Then replace A by A. , t h u s yielding an element al of A. satisfying the same condition and taken minimum. By i t e r a t i o n , we obtain an w-sequence o f elements ai ( i integer) , a l l of whose pairs of elements have color ( - ) . Assume now t h a t A = ud i s singular. Then c4 i s a lim it ordinal (ch.2 9 6.2, axiom of choice). Let UUc and o(. be the co f i n al i t y of un , SO Thus M, i s the ordinal l i mi t of the r-sequence w , where i < 2( m(i) and g ( i ) < o( Moreover, we can choose the LX ( i ) t o be s t r i c t l y increasing with i , and every o \ ( i ) > 8 Finally every w g ( i ) can be assumed t o be regular, replacing i f need be d ( i ) by i t s successor. Suppose t h a t the f i r s t conclusion f a i l s : there i s no denumerable subset o f A Then by the preceding, there e x i s t s a subset a l l of whose pairs have color ( - ) B of A , equipotent with A , such t h at f o r every x in B , there are many y in B with { x,y) having color ( - ) s t r i c t l y l e ss t h a n For every subset X of B , denote by M ( X ) the s e t of elements of B-X which, together w i t h a t l e a s t one element of X , have color ( - ) . Let U be any subset of B equipotent with A , and l e t i be an ordinal s t r i c t l y less than 8 . We shall show t h at there e x i s t s a subset W of U with cardinal IAM ( i ) ' satisfying the two following properties: every pair of elements of W has color (+) ; the s e t M ( W ) has cardinal s t r i c t l y l ess t h a n ma. Indeed, by our f i r s t paragraph and because f o r every i < 8 the cardinal L3 %(i) i s regular, there e x i s t s a subset V of U with cardinal w N c (i) , a l l of whose pairs have color (+) . For every j < d , denote by V the s e t of j elements x of V , such t h a t there e x i s t a t most woc(j) elements in B - i x ] which together with x have color ( - ) . Then V = \I V . ( j < b') since, J by our third paragraph, no x together with wd many elements of B , has color ( - ) , and i s the limit of the (j C I f ) Recall t h a t the

.

.

r<

.

x+

.

.

Q,

.

.

THEORY OF RELATIONS

72

.

c a r d i n a l wec(i) o f V i s r e g u l a r and s t r i c t l y g r e a t e r than It f o l l o w s t h a t there e x i s t s a t l e a s t one o r d i n a l k 6 8 w i t h Vk equipotent w i t h V : p u t

<

. Then Card M(W) ,< W,c(i). L3 o ( ( k ) do(. Thus the two p r o p e r t i e s s t a t e d above f o r W are obtained. It remains t o c o n s t r u c t a subset o f B which i s equipotent w i t h B and thus w i t h A , a l l o f whose p a i r s have c o l o r (+) . L e t W1 be a subset o f B w i t h cardinal w , a l l o f whose p a i r s have c o l o r (+) , w i t h Card M(W1) s t r i c t l y o((1) l e s s than ~3~ I t e r a t e by t a k i n g W2 a subset o f B (W1 u M(W1)) w i t h card i n a l G)o((2) , a l l o f whose p a i r s have c o l o r (+) , w i t h Card M(W2) s t r i c t l y

W = Vk

.

-

.

l e s s than ucANote t h a t , i n t h e union W1u W2 , a l l t h e p a i r s have c o l o r (+). L e t i < 8 and assume t h a t t h e Wi are defined f o r j < i Then the union

.

For (Wj u M(W.)) ( f o r a l l j < i ) has c a r d i n a l s t r i c t l y l e s s than a d . J otherwise t h e i-sequence o f c a r d i n a l s Max( W ,Card M(W.)) would y i e l d a d(j) J and Card M(Wj) < c 3 ~ f o r each j < i sum >, Wd , w i t h Cdo((j) J

.

<

Hence t h e c o f i n a l i t y o f wOc would be 6 i < 8 , c o n t r a d i c t i n g t h e f a c t t h a t F o r U take the difference B u ( W . u M(Wj) f o r 8 i s the c o f i n a l i t y o f J a l l j < i) : t h i s has c a r d i n a l i t y ad o f t h i s d i f f e r e n c e which s a t i s f i e s the two proHence t h e r e e x i s t s a subset Wi w i t h a l l p a i r s o f Wi having c o l o r (+) , and p e r t i e s : Card Wi = CJ M(i) Card M(Wi) I t remains t o note t h a t the union o f t h e s t r i c t l y l e s s than L,J~ 0 Wi ( i < 8 ) has c a r d i n a l W, and t h a t a l l i t s p a i r s have c o l o r ( t ) Example. For u 1 we o b t a i n e i t h e r a denumerable (-)-monochromatic subset, o r With t h e n o t a t i o n o f Ramsey a (+)-monochromatic subset having c a r d i n a l W l numbers: ( a, =

.

-

.

.

.

.

a1

-

3.4. PARTITION LEMMA (ERDOS 1942) Let

oa be

an i n f i n i t e aleph. Set

A

=

WM+2

and assume t h e generalized

.

i ) 2 = c3 i+l f o r every i.4< d Then e i t h e r P a r t i t i o n t h e p a i r s o f elements o f A i n t o two c o l o r s (+) and (-) 3 , o~r t h e+r e ~ there e x i s t s a (+)-monochromatic subset o f A w i t h c a r d i n a l ~ See E R D b 1942 o r e x i s t s a (-)-monochromatic subset w i t h c a r d i n a l w 0(+2 2 ERDOS, RADO 1956. With Ramsey numbers: ( w o(+l, %+2) = c30(+2 . 0 We prove f i r s t t h a t t h e r e e x i s t s a monochromatic subset o f A w i t h c a r d i n a l o(+l Take ‘4 = 0 , so t h a t A = W 2 : we s h a l l o b t a i n a monochromatic s e t w i t h c a r d i n a l CC, The p r o o f w i l l e a s i l y extend t o t h e general case. We say t h a t a sequence of terms ai i n A (io r d i n a l ) i s pre-monochromatic i f f for every index i , t h e c o l o r o f t h e p a i r s {ai,aj) remains t h e same f o r a l l j > i : we say t h a t t h e c o l o r i s associated t o t h e index i Construct as f o l l o w s continuum hypothesis i n t h e form

(

.

.

.

.

.

Chapter 3

73

a pre-monochromatic u l - s e q u e n c e . W e then immediately extract a monochromaGJ l-sequence, by taking a l l those indices i with the same color, provided t h i s s e t i s cofinal in CJ . a. denote the minimum of A ( i n the usual well-ordering of A = "2 ; thus a. = 0 , the value i s unimportant). Partition the elements x # a. of A into two classes: the class of those x such that { a o , x j has color (+), and the class 1 similarly defined with ( - ) . Let a; be the minimum in the class into two subclasses: the 8 , and then partition the elements x # a; of such tic that Let

df

i

1

ri

t'

class

i-

of those x for which { a l , x ) has color (+), and the class similarly defined with ( - ) . Similarly, l e t a; be the minimum of the class , and then p a r t i t i o n the elements of d i s t i n c t from t h i s minimum, into

8 ;+ and

two subclasses I n general, l e t

u

xi

r i-

, defined as previously. be a n ordinal s t r i c t l y less than w

.

If

u

has a prede-

d

cessor u-1 , assume t h a t the classes :-1 are already defined, each characterized by a sequence s of + and - , with length u-1 , hence defined on the indithe minimum of , provided t h i s ces s t r i c t l y l e s s t h a n u . Denote by a:-1 class i s non-empty. Partition the elements in t h i s class which are d i s t i n c t from the minimum (assuming of course t h a t there are such), into two subclasses: the

r:-l

class

$

:+,

characterized by the sequence

u , of

hence a sequence of length

gous definition f o r the class

1(

s completed by the

x f o r which {a:-l,x)

:-.

( ~ - 1 ) term '~ + ,

has color ( + ) . Analo-

Suppose now t h a t u i s a l i m i t ordinal. Given a sequence s of length u , hence of indices s t r i c t l y l e s s t h a n u , consider f o r each i < u the r e s t r i c t e d sequence s / i taking the same values as s , b u t defined only f o r indices < i Then we define the class as the intersection of the classes 8 :Ii f o r a l l

.

r:

ordinals

i

<

u

. Finally,

whether

u

i s a l i m i t ordinal o r n o t , we define the

element a: as the minimum of f: , provided t h a t t h i s c l a s s i s non-empty. Since the ordinals considered a r e a t most countable, there are continuum many sequences s , hence u 1 many, since we assume the continuum hypothesis. Hence there are cjl many classes 4 a n d t h e i r minimums a , f o r a l l indices u and a l l sequences s . Since A has cardinality &J2 , there e x i s t other elements besides the minimums a

.

Let r be one such. Beginning with a. , pick e i t h e r a; o r a; , depending on whether the pair \ a o , r ) has color (+) or ( - ) . I f we have chosen a: , then choose either u


a?

or

, if

a;by the same consideration. Continue thusly: given an ordinal we have already chosen the sequence s with length u-1 and values

THEORY OF RELATIONS

14 (+) and ( - ) , choose

a:'

, depending on whether the pair {a:-l,r}

or a:-

has

the color (+) o r ( - ) . For every l i m i t ordinal u , take the sequence s with length u which i s the l i m i t of the sequences already obtained, and take the corresponding a s . U By the definition of the classes , whenever we reach a term a: in the sequence defined by r , a l l following terms belong t o the same class 8 , i.e. t h a t which contains r . Hence f o r every v u a l l pairs { a u , a v } have the same color as { a " , r ) . Moreover, the class y through which we pass i s never empty, since r belongs t o i t . W e thus obtain a pre-monochromatic W1-sequence from which we e x t r a c t , as already explained, a monochromatic wl-sequence. 0

>

Let us now take u p the proof of our stated lemma. Assume t h a t the second conclusion does not hold, so t h a t every (-)-monochromatic subset of A has cardinality a t most W 1 . Take a subset Do of A , which i s maximal with respect t o inclusion among the . For every e l e (-)-monochromatic subsets; hence Do has cardinality ,< W ment x of A-Do , there e x i s t s a t l e a s t one element y of Do such t h a t the pair ( x , y ) has color ( t ) . Associate t o each x of A-Do the minimum such y . Thus we p a r t i t i o n the elements of A-Do into classes, each defined by an element a. of Do . W e denote by 'd (a,) the class thus associated with a, , with the color (+) f o r {ag,x) f o r a l l x in t h i s c l a s s . If If(a,) i s non-empty, then take a subset Dl(ao) of t h i s class which i s maximal with respect t o inclusion among (-)-monochromatic subsets; hence D1 has cardinality U l . As previousl y , p a r t i t i o n the elements of (a,) - Dl(ao) into classes which are defined by an element a l of Dl(aO) . W e denote by (a,,al) the class thus associated with the sequence (a,,al) , with the color (+) f o r { ao,x) and f o r { al,x) for every x in t h i s c l a s s . The i t e r a t i p n can be continued in an obvious manner f o r a l l successor ordinals u s t r i c t l y less t h a n cJ1 For a l i m i t ordinal u and a sequence s with length u , having values a i ( i < u) , we define the class 8 ( s ) as the intersection of the g ( s / i ) f o r a l l i < u ( r e c a l l t h a t s / i i s the r e s t r i c t i o n o f s t o indices s t r i c t l y l e s s than i ) . Of course, each such sequence s = ao,al, ...,a i ' " must s a t i s f y the following conditions. For every successor ordinal i < u , the term a i belongs t o Di(ao,al,.. . , a i - l ) , which i s a maximal (-)-monochromatic subset of r i ( a D , a l ,...,a i - l ) For every l i m i t ordinal i < u , the term ai belongs t o Di(ao,a l , . . . , a j , . . ) ( j < i ) , which i s a maximal (-)-monochromatic

0

.

.

subset of

r i ( a o , al , . . . , a j , . . )

, the l a t t e r being the intersection of the

.

8 j ( a o , a l , . . . , a j - l ) (j successor ordinal < i ) Notice t h a t the axiom of choice D : we do not have i n i t i a l l y a well-ordering of the s e t o f i s used t o define subsets of A .

75

Chapter 3


u

For e v e r y o r d i n a l

, there

a r e a t most & ( U 1)

many sequences

...

w i t h l e n g t h u , t h u s a t most W 1 many, s i n c e we assume t h e continuum ao,al, h y p o t h e s i s . Hence t h e s e t o f a l l t h e a has c a r d i n a l i t y W 1 and l i k e w i s e f o r the union o f the belong t o no Let

.

D

n o t belong t o f o r every

quence o f these

a.

, pick

Do


u

A

such t h a t

al

pick

au

a l l contain

A

r

. Then

belongs t o

r

as

does

; and i n general,

. The

Z f ( a ,,...,au)

,

se-

s i n c e t h e above

RADO 1953)

<

an a l e p h

B

.

A

Partition the pairs o f

i n t o two c o l o r s (+) and ( - ) . Then e i t h e r t h e r e e x i s t s a subset of and ( + ) - m o n o c h r o m a t i c , z t h e r e -

A

potent w i t h

(-)-monochromatic

a

r(ao)

as an element, and so a r e a l l non-empty. 0

r

A equipotent w i t h (2) Let

e x i s t elements which

8 (ao,al)

belongs t o

such t h a t

be a r e g u l a r l i m i t aleph,

elements o f

belongs t o

r

r

such t h a t

PARTITION THEOREM (ERDOS, A

, there

W2

i s (+)-monochromatic and has l e n g t h (,d1

au

considered c l a s s e s

(1) L e t

has c a r d i n a l i t y

.

D

be one such. P i c k

r

3.5.

Since

B

be an i n f i n i t e aleph,

b

e x i s t s a subset o f

w-

A

(uses g e n e r a l i z e d continuum h y p o t h e s i s ) .

t h e l e a s t aleph s a t i s f y i n g

ba

>

a

. Let

A be t h e a l e p h a+ , t h e successor o f a . P a r t i t i o n t h e p a i r s o f elements o f A i n t o two c o l o r s (+) and ( - ) . Then e i t h e r t h e r e e x i s t s a subset o f A e q u i p o t e n t with A (+)-monochromatic, t h e r e exl’sts a subset of A e q u i p o t e n t w i t h

- and b

and (-)-monochromatic

(uses axiom o f c h o i c e ) .

0 (1) Assume t h a t e v e r y (+)-monochromatic

s h a l l c o n s t r u c t a subset o f

A

A

subset o f

equipotent w i t h

<

has c a r d i n a l i t y

A

. We

and (-)-monochromatic.

B

X o f A , a s s o c i a t e a subset T(X) o f X which i s (+)-monochromatic and maximal w i t h r e s p e c t t o i n c l u s i o n , among (+)-monochromatic

To e v e r y non-empty subset subsets o f worst

X

T(X)

(uses axiom o f c h o i c e ) . N o t i c e t h a t

For e v e r y element ment set

fi(x) fo(x)

Now l e t j

x

A

of

of

A

. Fix

< i

i

and e v e r y o r d i n a l

x ; take

fo(x) = x

t o be t h e l e a s t element i n

{

A ) , f o r which t h e p a i r

of

x,fo(x))

T(A)

if

Either there already e x i s t s a

case, s e t

i s non-empty: a t t h e

fi(x)

,Or

= x

a l l the

, we x

d e f i n e as f o l l o w s t h e e l e belongs t o

T(A)

.

has c o l o r ( - ) (uses m a x i m a l i t y o f j

J

f.(x)

J i f o r which

<

f.(x)

Otherwise

( w i t h respect t o the well-ordering

i be a non-zero o r d i n a l , and assume t h a t

.

T(X)

c o u l d be a s i n g l e t o n .

T(A) ) .

i s d e f i n e d f o r each f . ( x ) = x and i n t h i s J x and d i s t i n c t among

are d i s t i n c t from

themselves, and t h e p a i r s t h e y f o r m among themselves o r w i t h c o l o r ( - ) . Then l e t color (-) f o r a l l

j

, then

fi(x)

T(U)

set

maximality o f

T(U)

U

be t h e s e t o f t h e

<

i

.

= x

, there

I n particular

.

x a l l have t h e A f o r which ( y , f j ( x ) ] has belongs t o U I f x belongs t o in

y

.

x

Otherwise, i f

x

belongs t o

e x i s t elements

z

of

T(U)

U-T(U)

,

w i t h {x,z}

t h e n by t h e having

THEORY OF RELATIONS

16 c o l o r ( - ) . Take

fi(x)

z

t o be t h e l e a s t such

.

A

i n the well-ordering o f

x o f A , t h e r e e x i s t s an o r d i n a l ix ix, t h e f i ( x ) a r e d i s t i n c t f r o m x and d i s t i n c t f r o m each o t h e r , and t h e p a i r s t h e y f o r m among themselves o r w i t h x have c o l o r ( - ) ; and f i ( x ) = x f o r i 3 i x ' For e v e r y o r d i n a l i , l e t Mi be t h e s e t o f these f i ( x ) , w i t h i f i x e d and x r u n n i n g through A . We s h a l l p r o v e t h a t Mi has c a r d i n a l i t y < A f o r e v e r y By t h e p r e c e d i n g c o n s t r u c t i o n , f o r e v e r y

such t h a t , f o r a l l

i

<

i

. First of

A

<

all,

Mo

i s contained i n

<

hence by h y p o t h e s i s has c a r d i n a l i t y

me t h a t f o r e v e r y i

-= C

we have

.

A

Card Mi

T(A)

which i s (+)-monochromatic,

Consider an o r d i n a l

<

.

A

Since

A


C

and assu-

i s r e g u l a r by hypo-

Mi ( i < C ) i s A . Otherwise A would be t h e u n i o n o f s t r i c t l y < A many s e t s , each s t r i c t l y subpotent w i t h A L e t D denote t h e maximum o r supremum c a r d i n a l o f t h e Mi . To e v e r y element x o f A , a s s o c i a t e t h e sequence o f t h e

t h e s i s , t h e maximum c a r d i n a l o r t h e supremum c a r d i n a l o f t h e s t r i c t l y less than

.

fi(x)

. The

( i 4 C)

. Now by

(Card

number o f d i s t i n c t such sequences i s l e s s t h a n o r equal t o

the f a c t that

A

5

i s a l i m i t aleph, and by ch.2

r a l i z e d continuum h y p o t h e s i s ) , t h i s c a r d i n a l i s s t r i c t l y l e s s t h a n

6.5 ( 2 ) (gene-

.

A

Two i d e n t i -

U , hence t h e same T(U) o f c a r d i n a l i t y < A . For any x , t h e new element f C ( x ) belongs t o T(U) . Hence as x v a r i e s , t h e

c a l sequences g i v e t h e same s e t p o s s i b l e sequences regularity o f

A

By h y p o t h e s i s

B

. For

ir >/ B

fi(x)

, the

( i d C)

set

<

MB

A

.

A

A

o f cardinality

<

has c a r d i n a l i t y

with cardinality

b

.

a

many s e t s fC(x)

would be i d e n t i c a l w i t h

fi(r) a r e d i s t i n c t f o r c h r o m a t i c and o f c a r d i n a l i t y B . 0 (2) For

A

a+

, suppose

i

<

B

, and

. Because o f

T(U)

r

in

A

such t h a t

, contradicting

A

the

.

has c a r d i n a l i t y C A

Thus t h e r e e x i s t s an

ceding. Hence t h e

0

<

MC o f a l l p o s s i b l e

i s a cardinal

otherwise,

give

t h e pre-

t h e i r s e t i s (-)-monochro-

t h a t e v e r y (+)-monochromatic subset of

We s h a l l c o n s t r u c t a (-)-monochromatic subset o f

A

.

x o f A , c o n s t r u c t as p r e v i o u s l y t h e f i ( x ) and t h e Mi f o r a l l o r i (axiom o f c h o i c e ) . We s h a l l p r o v e t h a t , f o r e v e r y i o f c a r d i n a l i t y b , hence f o r e v e r y i < b , t h e s e t Mi has c a r d i n a l i t y 6 a Notice

For e v e r y dinals

<

.

f i r s t that

b

6

a

i s (+)-monochromatic hence

c

<

a

, and

>

since

.

a

The

set

and hence has c a r d i n a l i t y assume t h a t

o r t h e supremum o f t h e

Card Mi

Card Mi

(i

i s contained i n

6

a

. Take

c)

is

a a

. The

since

Card c

x

running through

Since

b

,< a , t h e

A

,

is

,<

union o f the

. (i<

c

. The

<

< b . As Mc

,

b

maximum

fi ( x ) (i < c)

sequences

p r o o f o f (1) above, i t f o l l o w s t h a t t h e c a r d i n a l i t y o f t h e s e t

, which

T(A)

an o r d i n a l

6 a f o r every i < c

, hence 6

(Card ')a

have c a r d i n a l i t y a t most

<

Mo

of

i n the

fc(x)

for

a.a = a Mi

b)

has c a r d i n a l i t y

6

b.a = a

.

Let

Chapter 3

77

r .be an element of A not in this union. Then r # fi(x) for every x of A and every i < b . In particular r # fi(r) for every i < b : thus i, 3 b . It follows that the set of the fi(r) (i < b) is (-)-monochromatic and has cardinality b . 0 Note that, in the proof of ( 2 ) , the cardinal A = a, is not a limit aleph. This condition, and also the use of generalized continuum hypothesis, is only necessary for (1) in order to apply ch.2 5 6.5.(2). The example of w1 , given at the end of 5 3.3, can be expressed here by taking a = d , b = w and A = b J 1 . Recall that with generalized continuum hypothesis, the only regular limit alephs are CJ and the inaccessible alephs (ch.2 5 6.7). The statement (1) holds only for these. On the other hand, with only the axiom of choice, the continuum can be regular or not, a limit or a successor aleph, with cofinality >/a(ch.2 § 6.4). §

4 - COMBINATORIAL LEMMAS,

COLOR AND INCLUSION

4.1. (1) Let E be a set, p, q two integers such that ptq 4 Card E . Take a set of p-element subsets included in E and call this the color u . If every (p+q)-element subset includes the same number k f p-element subsets with color & , then every (ptqt1)-element subset includes the same number with color k(p+qtl)/(qtl) of p-e&%n_t_Aybsets (2) Given two not necessarily disjoint sets of p-element subsets o f E , call these the colors ?d, and 2'. If every (p+q)-element subset includes as many p-element subsets with color p-element subsets with color ?f, then the same is true fmeve_ry (p+qtl)element subset of E (1) Let F be a (p+q+l)-element set. The cardinality of the set o f (p+q)element subsets of F is ptq+l Each includes k many p-element subsets with color k , which yields k(p+qtl) ordered pairs, each formed with a p-element set having color ?,(, and with a (ptq)-element set which includes it. For every p-element subset with color % in F , there are qtl many (ptq)as the number element subsets which include it. This yields k(p+q+l)/(q+l) of p-element subsets included in F and having color & . 0 (2) Let F be a (p+q+l)-element set. Every (p+q)-element subset o f F Thus we includes as many p-element subsets with color % as with color have that in F , there are the same number of ordered pairs, each with first term a p-element set with color 21 and second term a (p+q)-element set including the first term, as of ordered pairs, each with first term a p-element

u.

u

~

as

.

.

r.

THEORY OF RELATIONS

78

set having color 'It and second term a (p+q)-element set including the first term. We obtain the number of p-element subsets with color % by dividing the preceding number by q+l . Similarly for the color 'v . 0 4.2. (1) Let E be a finite set, p. q two integers such that p+q 6 Card E . Let 'LL be a color of certain p-element subsets of E . Let ss p and s 6 (Card E) - p - q . If every (ptq)-element subset includes the same number of p-element subsets , then every s-element subset is included in the same number of_ with color p-element subsets with color /d~. (2) Let %, be two colors of p-element sets, not necessarily disjoint. If every (p+q)-element set includes as many p-element subsets with color % as p-element subsets with color 1/, then every s-element subset is included in as many p-element sets with color 2 p-element sets with color (communicated by POUZET in 1975). 0 First we prove (2). For s = 0 , this follows from 4.1.(2) iterated from p+q to Card E . Assume that s 31 and assume that the statement is true for s - 1 and p+q . In other words, for every E with finite cardinality >/ s+p+q-1 and every (s-1)-element subset of E . We shall prove this for s and p+q , hence for E with finite cardinality 3 s+p+q and a s-element subset H G E . Let u be an element of H . By the induction hyoothesis, there exists a same number k of p-element subsets with color % as with color 2 / , included in E and including H - {u) . Similarly, there exists a same number 1 of p-element subsets with color as with color included in E - { u) and including H - {u} (the cardinality of these sets being respectively s+p+q-1 and equal to s-1 ). By subtraction, there existsthesame number k-1 of p-element subsets with color ?A,'! as with color , included in E and including H .O 0 Statement (1) follows from (2). Indeed, let H and H' be two s-element subsets of E . Take a permutation f of E which transforms H into H ' . Take the p-element subsets with color , and let 21' be the color of their images via f . Then every (p+q)-element subset X includes as many p-eleas one can see by taking the ment subsets with color as with color (p+q)element subset (f-l)"(X) (notation from ch.1 !j 1.2). By ( 2 ) , the s-element subset H ' is included in as many p-element subsets with color %& as with color . But the latter are the images via f of p-element subsets with and including H . Thus H and H ' are included in the same number color of p-element subsets with color u . 0

u

u

u

r,

u

u

u

u'

a',

Chapter 3

79

4.3. (1) Let E be a set, p an integer less than or equal to Card E , and let be a non-empty set, called color, of p-element subsets.

u

such that 2p + q Card E , and if every (p+q)element set includes the same number of p-cgment subsets with color u , then every p-element subset has color ld . (2) Given E and p less than or equal to Card E , let I d , 3‘ be two sets, called colors, of p-element subsets. If there exists an integer q such that 2p + q 4 Card E , and for which every (p+q)-element subset includes as many p-element subsets with color p-element subsets with color 1;’ , then the colors and 2/ are identical. I f there exists an integer q

_

_

~

_

^

_

_

can assume that E is finite, by replacing E if necessary by a finite subset of cardinality at least equal to 2p + q . Now take the preceding statements with s = p . By statement (l), every p-element subset is included in the same number of p-element subsets with color 16. In other words, every p-element subset has color 1 1 (since it is assumed that 2 is non-empty). By statement (2), every p-element subset is included in as many p-element subsets with color 2 as p-element subsets with color 17. In other words, the are identical. 0 colors ‘lG and

0 We

If Card E < 2p + q , then by taking s 6 (Card E)-p-q , it is easy to give an example in which the color does not extend to the entire set of p-element subsets. Thus with E = {a,b,c,dj hence of cardinality 4 , with p = 2 , q = 1 , and only the edges ab and cd with color % , every 3-element subset contains such an edge, and every element belongs to such an edge. Adding ac and bd with color every 3-element subset contains an edge with each color, and all elements belong to an edge of each color.

v,

Calculation. If e designates the cardinality of E , and k the number of p-element subsets of color u? contained in every (p+q)-element subset, then by 4.1.(1 , the number of all p-element subsefis with color a is k.e!q!/(p+q)!(e-p)! . The number of those which contain a given element u , hence are not contained in the (e-1)-element subset E - (u) , is obtained by subtraction. It equals / (p+q)!(e-p)! , by assuming e >/ p+q+l . The number of those which k.(e-l)!q!p , hence which contain u and are not contained in contain a given pair (u,v) the (e-1)-element subset E - { v ) , i s obtained by subtraction. It equals , in assuming e >/ p+q+2 . In general, the numk.(e-2)!q!p.(p-1) / (p+q)!(e-p)! ber of those containing a given s-element subset is: , in assuming s & e-p-q . k.(e-s)!q!p! / (p+q)!(e-p)!(p-s)!

THEORY OF RELATIONS

80

§

5 - INCIDENCEMATRIX, MULTICOLOR THEOREM

KANTOR'S

L I N E A R I N D E P E N D E N C E LEMMA,

5.1. INCIDENCE MATRIX Let p. q be two integers, E a s e t of f i n i t e cardinal h Represent as "ordinate values" the s e t of p-element subsets of E , of cardinality h!/p!(h-p)!, and as "abscissa values" the s e t of (p+q)-element subsets, of cardinality To each couple ( x , y ) where x i s a (p+q)-element h!/(p+q)!(h-p-q)! s e t and y i s a p-element s e t , a t t r i b u t e the value 1 i f f y s x and the value 0 otherwise. The rectangular t ab l e thus obtained shall be called the incidence matrix of E f o r p and q Note t h a t i f h = Card E 2 2p + q , then each row of the matrix, corresponding t o a p-element s e t , i s a t l e a s t as long as each column, corresponding t o a (p+q)-element s e t : indeed p!(h-p)! 3 ( p + q ) ! ( h - p - q ) ! The reader i s assumed t o be familiar with the elementary theory of determinants and with the notion of l i n ear dependence. If Card E < 2p + q , then i t i s possible t h a t a row of the incidence matrix depends linearly on one or several other rows. For example, f o r p = q = 1 and Card E = 2 , the matrix reduces t o two rows and one column, with value 1 . B u t f o r Card E 3 2 p + q , we have the following r e s u l t . LINEAR INDEPENDENCE LEMMA I f Card E 3 2p + q , the rows of the incidence matrix a re linearly independent: no row i s a l i n e a r combination of other rows. Equivalently, every non-zero determinant extracted from the matrix and depending on a f i n i t e number r of rows can be extended t o a non-zero determinant based on the previous rows together with an a r b i t r ar y (r+l)st row. I n the case t h a t E i s f i n i t e , i t follows t h a t there e xists a non-zero determin a n t depending on a l l the rows. Hence there ex i s t s an~injection _ _ which _ t o each p-element s e t y associates a (p+q)-element s e t including y as a subset (KANTOR 1972). 0 To each permutation f of E associate the corresponding permutation f " of Hence f " permutes the s e t of rows. There corresponds p-element subsets of E t o f as well a permutation of the s e t of (p+q)-element subsets, hence of the s e t of columns, b u t i t i s unnecessary t o consider t h i s , since we are working with l i n e a r combinations of rows and reasoning by the coefficients a ttribute d t o each row in a given l i n ear combination. Assume t h a t E has f i n i t e cardinal h ; we argue ad absurdum. Assume t h a t there e x is t s a p-element subset, hence a row which i s a line a r combination of a l l the other rows, with positive, negative or zero rational coefficients, since these are quotients of determinants with values 0 or 1

.

.

.

-

.

~

.

.

Chapter 3

L e t us c a l l

b this

81

p-element s e t and t h e corresponding row. Given an a r b i t r a r y

permutation f o f E which preserves the s e t b ( b u t not n e c e s s a r i l y each element o f b ), then f o preserves t h e row b and permutes t h e s e t o f t h e other rows; two rows which are transformed one i n t o t h e o t h e r represent two p-element sets y, y ' such t h a t b n y and b A y ' are equipotent. Transform the given l i n e a r combination by a l l p o s s i b l e fa

, the

number o f such

being (h-p)!p! , then take t h e combination which i s t h e a r i t h m e t i c average o f the combinations thus transformed. By s y n e t r y , a l l t h e rows which represent p-element sets d i s j o i n t from b w i l l have t h e same c o e f f i c i e n t . S i m i l a r l y f o r a l l rows which represent p-element s e t s having a s i n g l e element i n common w i t h b , and i n general f o r a l l rows which represent p-element sets having equipotent intersections with b Consider a column a representing a (p+q)-element s e t d i s j o i n t from b : t h i s

.

.

I n the column a , t h e p-element sets included a e x i s t s since h >/ 2p + q i n a a r e a l l d i s j o i n t from b , and so a l l have t h e same c o e f f i c i e n t i n o u r combination. Moreover, i f we denote these p-element sets by y , these are the i n t h e incidence matrix, o n l y ones y i e l d i n g t h e value 1 i n p o s i t i o n (a,y) w h i l e t h e m a t r i x has t h e value 0 i n p o s i t i o n (a,b) It f o l l o w s t h a t t h e i r

.

c o e f f i c i e n t i s zero, hence each row which represents a p-element s e t d i s j o i n t from b has c o e f f i c i e n t zero. The problem i s thus answered n e g a t i v e l y f o r p = 1 , since i n t h i s case the p-element sets d i s t i n c t from b are d i s j o i n t w i t h b , hence t h e above assumed l i n e a r combination does n o t e x i s t . 2 , and consider a column al representing a (ptq)-element Assume t h a t p s e t which i n t e r s e c t s b i n a unique element. Then t h e rows y f o r which t h e m a t r i x has value 1 i n (a,,y) a r e those which represen! e i t h e r a p-element s e t d i s j o i n t from b , hence w i t h c o e f f i c i e n t zero, o r a p-element s e t i n t e r s e c t i n g b i n a s i n g l e p o i n t . By t h e preceding discussion, t h e l a t t e r have t h e same c o e f f i c i e n t i n the combination. Since the m a t r i x has the value 0 i n (al,b) , t h i s c o e f f i c i e n t i s zero. The problem i s thus answered n e g a t i v e l y f o r p = 2 , s i n c e i n t h i s case t h e p-element s e t s d i s t i n c t from b have a t most one element i n common w i t h

b

. I n the

general case, by i t e r a t i n g t h e preceding

argument, we prove t h a t a l l t h e c o e f f i c i e n t s are zero, .hence t h a t t h e above assumed l i n e a r combination does n o t e x i s t . The r e s u l t f o l l o w s imnediately i n t h e case o f E i n f i n i t e . F i n a l l y , f o r t h e cnnclusion concerning t h e e x t e n d i b i l i t y o f a non-zero determinant, assume on t h e c o n t r a r y t h a t t h e r e e x i s t s a non-zero determinant which i s n o t extendible, and deduce t h a t an a r b i t r a r y row o f t h e m a t r i x i s a l i n e a r combination o f rows o f the submatrix which corresponds t o t h i s determinant,

82

THEORY OF RELATIONS

5.2. In t h e "degenerate case" where h = Card E < 2p+q , t h e number of columns i s s t r i c t l y l e s s than t h e number of rows. In t h i s case t h e columns of t h e incidence matrix a r e l i n e a r l y independent; i n o t h e r words, t h e r e e x i s t s a non-zero determinant based on t h e col umns. 0 Interchange each p-element s e t y w i t h t h e (h-p)-element s e t E-y , and each (p+q)-element s e t x with t h e (h-p-q)-element s e t E-x . Then the inclusion E-y . The r o l e of p i s played by p ' = h-p-q ; y c x i s equivalent t o t h e r o l e of p+q i s played by p ' + q ' = h-p , so t h a t q ' = q . We have 2p' + q ' = 2 h - 2 p - q < h : hence we can apply t h e l i n e a r independence lemna with rows and columns interchanged. 0

E-xc

5.3. MULTICOLOR Let E be a f i n i t e s e t , h i t s cardinal and p , q two i n t e g e r s . P a r t i t i o n the p-element subsets of E i n t o a f i n i t e number k of c l a s s e s which a r e c a l l e d colors uo, ul, ..., u ~ .- For ~ each (p+q)-element subset a of E , we c a l l t h e multicolor of a t h e function which t o each c o l o r u i ( i k ) associates t h e number of p-element s e t s of c o l o r u i which a r e included i n a When t h i s number i s non-zero, we say t h a t t h e c o l o r u i f i g u r e s in the multicolor. MULTICOLOR THEOREM I f Card E 2, 2p+q , then t h e number of m u l t i c o l o r s ' o f (p+q)-element subsets

.

of

E i s a t l e a s t equal t o t h e number of c o l o r s of p-element subsets. More p r e c i s e l y , t h e r e e x i s t s an i n j e c t i o n which t o each c o l o r u ( t o which a t l e a s t one p-element s e t belongs) a s s o c i a t e s a multicolor i n which u f i g u r e s , and t o which a t l e a s t one (p+q)-element s e t belongs (POUZET 1976). Assume f i r s t t h a t E has f i n i t e cardinal h > , 2p+q . Hence t h e number of (p+q)-element s e t s i s a t l e a s t equal t o t h a t of t h e p-element s e t s , and the rows of t h e incidence matrix a r e l i n e a r l y independent. To each c o l o r t h e r e corresponds a f i n i t e s e t of rows of t h a t c o l o r . Replace these by a unique row which i s t h e i r sum, obtained by adding the values 0 o r 1 i n each column. T h u s each new row represents a c o l o r u Each column continues t o represent a (p+q)-element s e t , and i n d i c a t e s the number of p-element sets of c o l o r u which a r e included i n t h i s (p+q)-element set. Note t h a t , i n t h e new matrix t h u s obtained, t h e rows a r e l i n e a r l y independent. I t s u f f i c e s t o see t h a t , given a matrix w i t h k independent rows ( k 3 2 ) , the replacement of two rows b and b ' by t h e i r sum y i e l d s a matrix w i t h k-I independent rows. Indeed, t h e r e e x i s t s a non-zero determinant based on the k-2 i n t a c t rows. So t h a t the only o t h e r p o s s i b i l i t y would be t h a t t h e row sum o f b and b' i s a l i n e a r combination of the k-2 i n t a c t rows. B u t then the row b , f o r example, would be a l i n e a r combination o f t h e k-2 i n t a c t rows plus

.

83

Chapter 3

the row b' , c o n t r a d i c t i n g t h e hypothesis. T h u s , i f k i s now t h e number of c o l o r s , hence of rows, we have a non-zero determinant of order k . Take i n this determinant a sequence of k ordered p a i r s ( x , y ) where x i s a column and y a row, w i t h non-zero value of the new matrix i n each considered ordered p a i r . We thus o b t a i n the i n j e c t i v e function i n the theorem. This i n j e c t i o n a s s o c i a t e s , t o two d i s t i n c t c o l o r s y. y ' two ( p + q ) element s e t s x , x ' whose m u l t i c o l o r s a r e d i s t i n c t . Otherwise we would have two i d e n t i c a l columns i n t h e determinant. Thus t h i s is an i n j e c t i o n from t h e s e t of colors i n t o the s e t of m u l t i c o l o r s . I t remains t o consider t h e case when E i s countably i n f i n i t e . I f we only have a f i n i t e number of c o l o r s , then we r e s t r i c t E t o a s e t of f i n i t e c a r d i n a l i t y a t l e a s t equal t o 2p+q and including p-element s u b s e t s of each c o l o r . The rows, which r e p r e s e n t t h e c o l o r s , a r e l i n e a r l y independent, and remain so when one takes up the e n t i r e i n f i n i t e set E . I f t h e r e a r e i n f i n i t e l y many c o l o r s , then we s t i l l have l i n e a r independence. Then a s mentioned f o r t h e l i n e a r independence lennna, every non-zero determinant i s extendible t o a non-zero determinant over one more row, hence one more c o l o r . The e x i s t e n c e of t h e i n j e c t i v e function i n the theorem follows. 0

5 6 - RAMSEY

SEQUENCE: ANOTHER PROOF OF

GALVIN'S

THEOREM

The following notion of Ramsey sequence of conditions is a form of the c l a s s i c a l Ramsey s e t : see ERDOS, RADO 1952. The connected proof of GALVIN's i n i t i a l i n t e r v a l theorem i s due t o LOPEZ 1983'. As opposed w i t h POUZET's proof i n 5 2 , here we need n e i t h e r lexicographic rank nor t r a n s f i n i t e induction. As well a s i n 5 2 , t h e axioms of ZF w i l l be s u f f i c i e n t : see 5 6.5 below.

6.1. Given two s e t s A, B of i n t e g e r s , p u t A < B o r B > A i f f every element of B i s s t r i c t l y g r e a t e r than every element of A . We adopt the convention t h a t the empty s e t i s < and > any s e t ; so t h a t < i s i r r e f l e x i v e and t r a n s i t i v e only f o r non-empty s e t s . L e t H be a f i n i t e s e t , Z an i n f i n i t e set of i n t e g e r s . A f i n i t e sequence of conditions g i ( H , Z ) ( i = 1, ..., r ) i s s a i d t o be a Ramsey sequence i f f we have the following: V H fin i n f X > H 93, i n f YCX A ( b z i n f Z c y 3rl(H,Z))V ...

VX

[

... v ( d zi n f

Z s Y

+Vr(H,Z))]

( n o t a t i o n s : f i n = f i n i t e , i n f = i n f i n i t e set of i n t e g e r s ; obvious l o g i c a l symbols).

THEORY OF RELATIONS

84

Example. P a r t i t i o n t h e p a i r s o f i n t e g e r s i n t o two c o l o r s (+) and (-). Take f o r

i n H , a l l p a i r s 1h.z) h ) " . Then alone

f(H,Z) t h e f o l l o w i n g statement: " f o r each i n t e g e r h where z belongs t o Z , have same c o l o r (depending on

,

c o n s t i t u t e s a Ramsey sequence. Another example. Take a c o n d i t i o n f and d e f i n e i g as t h e negation o f d i s o f t e n a Ramsey sequence. I t i s t h e case, for Then t h e sequence ( ff, -I instance, i f yf (H,Z) means t h a t the preceding p a i r s { h,z) have c o l o r (t). I n the case o f two such opposite conditions, the above formula means t h a t , given

.

e)

H , the s e t o f a l l i n f i n i t e Z s a t i s f y i n g C(H,Z) i s Ramsey i n the sense o f ERDOS, RADO 1952. I n other words, there e x i s t s an i n f i n i t e s e t Y o f i n t e g e r s such t h a t e i t h e r each i n f i n i t e Z C_ Y belongs t o t h e s e t defined by and H o r each i n f i n i t e Z c Y belongs t o t h e complement. Among sets o f i n f i n i t e sets o f integers, i . e . among sets o f r e a l s , i t i s known t h a t t h e f o l l o w i n g a r e Ramsey: a l l open sets (see t h e topology defined i n ch.1 exerc. 4 ) ; Bore1 sets (see GALVIN, PRIKRY 1973); a n a l y t i c s e t s (SILVER 1970). See a l s o ELLENTUCK 1974, who characterizes t h e "completely Ramsey sets" by t h e Bai r e property.

el ,..., er , we have t h e f o l l o w i n g

6.2. Given a Ramsey sequence

statement,

modulo t h e axiom o f dependent choice:

J ~ i ~ f f l ~ f i(vzinf ,, (

...

(Vz

inf

H C A A Z S A ~ Z > H ) V~1 ( H J ) ) V - - *

( H c A h Z E A A Z > H ) 3 f,(H,Z))

0 The p r o o f generalizes the f i r s t p a r t o f RAMSEY's p r o o f 1 . 1 , i n o b t a i n i n g e l e -

.

.

S t a r t from uo = 0 , Ho = { D l and Xo = s e t o f i n t e g e r s # 0 We ments vi get an i n f i n i t e Ys Xo , c a l l e d Yo and s a t i s f y i n g t h e above c o n d i t i o n i n

.

and A = {O)uYa Then l e t u1 be t h e f i r s t element { uo,ul) and XI = Yo -{ul) We g e t an i n f i n i t e

brackets, where H = Ho of Yo S t a r t from HI

.

.

=

Y1s X1

which s a t i s f i e s our above c o n d i t i o n i n brackets, where H = and A = \uo,uI) u Y1 Then s t a r t from H i = {ul) and = Y1 infinite Y1 which s a t i s f i e s our condition, where H = Ho o r and A = {uo,ul)u Yi Then l e t u2 be t h e f i r s t element o f Y i

Yi

.

.

Xi

.

Yi

and X2 = - {u2). from H2 = {uo,u1,u2) s a t i s f i e s our condition, where H = Ho o r

.

Ho

or

H1 an or H i

. We g e t HI Start

We g e t an i n f i n i t e Y2 5 Y i HI o r H i o r H2 and where

which

A = {uo,u1,u2\ u Y2 I t e r a t e , t a k i n g f o r H i , H;,.. a l l the sets w i t h l a s t element u2 , and so g e t t i n g Y h y Y i , before d e f i n i n g u3, Hg and Y3 ; and so on. F i n a l l y take f o r A t h e s e t o f ui ( i i n t e g e r ) The axiom o f dependent choice i s used f o r choosing s e t s Y . 0

...

.

6.3. L e t H, F be f i n i t e sets o f i n t e g e r s and Then t h e p a i r o f t h e f o l l o w i n g c o n d i t i o n e(H,Z) (HJ) c ( H , Z ) : 3 F fin F c Z A

a

.

b (H,F)

be an a r b i t r a r y c o n d i t i o n .

w i t h i t s negation i s Ramsec

Chapter 3

85

0 Suppose the contrary. There e x i s t a f i n i t e s e t

H

and an i n f i n i t e s e t

X 7 H

such t h a t , f o r every i n f i n i t e s e t Y 5 X , there e x i s t two i n f i n i t e subsets Z1 and Z2 with (H,Z1) and the negation iff (H,Z2) Then each f i n i t e subset F of Z2 s a t i s f i e s (H,F) Now replace Y by Z2 : there e x i s t s an i n f i n i t e subset Z i of Z2 such t h a t 'f: ( H , Z i ) . Thus there e x i s t s a f i n i t e subset F of (H,F) : contradiction. 0 5 Z2 which s a t i s f i e s

.

.

a

Zi

6 . 4 . A PROOF OF GALVIN'S INITIAL INTERVAL THEOREM be a s e t of f i n i t e s e t s of integers, assumed t o be mutually incomparable Let under inclusion, and t o s a t i s f y GALVIN's hypothesis: every i n f i n i t e s e t of integers includes a t l e a s t one element of 3 as a subset. Take 3 (H,F) t o be the following condition: "the union H u F admits an i n i t i a l interval belonging t o " ; more b r i e f l y " H,J F has i . i . " . Then by 6 . 2 there e x i s t s an i n f i n i t e s e t A such t h a t , f o r every f i n i t e subset H of A : either (1) inf ( Z g A A Z > H ) =) 3, f i n ( F c Z A H u F has F i . i . )

v,

or ( 2 )

tJz

inf ( Z 5 A

A

Z 7 H) =)

vF f i n

( F c Z

+

HuF

has no

'3 i . i . ) .

F i r s t l y we prove t h a t , assuming GALVIN's hypothesis, there e x i s t s an i n f i n i t e s e t A such the above ( 2 ) i s f a l s e : so only (1) i s true. 0 For H empty, the above conclusion ( 2 ) i s f a l s e . Indeed ( 2 ) reduces t o saying t h a t , f o r every i n f i n i t e s e t Z , there does n o t e x i s t any f i n i t e subset of Z which belongs t o Now l e t G be a f i n i t e s e t of integers. Assume t h a t the above ( 2 ) i s f a l s e f o r every subset H of G . Then i t suffices t o prove t h a t there e x i s t s an integer g 7 M a x G such t h a t every H 5 G u{g> f a l s i f i e s ( 2 ) . For t h i s purpose, i t suffices t o prove t h a t , f o r each subset H of G , there e x i s t only f i n i t e l y many integers h Z Max G such t h a t H u { h ) s a t i s f i e s ( 2 ) : indeed i t will suffice t o choose g s t r i c t l y greater t h a n a l l such h Arguing ad absurdurn, assume the existence of an i n f i n i t e sequence h l , h 2 , ..., h i , ... with Hu{hi} satisfying ( 2 ) . Take Z t o be the i n f i n i t e s e t of these h i : by hypothesis H f a l s i f i e s ( 2 ) , so H v e r i f i e s (1). Thus there e x i s t s a f i n i t e subset F of Z such t h a t H u F has an i n i t i a l interval which belongs t o 3-. Let h P (p integer) be the f i r s t element of F ; then H w{h ) f a l s i f i e s (2): contraP diction. 0

-

.

.

Secondly we obtain GALVIN's conclusion. A of 6.2, now assumed t o s a t i s f y only the above ( 1 ) . Let B be an a r b i t r a r y i n f i n i t e subset of A Let K be a f i n i t e subset of B belonging t o , and denote by H the i n i t i a l interval of B which ends with Max K . Then by ( 1 ) above, there e x i s t s a f i n i t e subset F of 6-H such

0 Consider the i n f i n i t e s e t

.

THEORY OF RELATIONS

86

.

t h a t HuF has an i n i t i a l interval which belongs t o 3 This i n i t i a l interval are mutually incomparable under cannot surpass Max K , since elements of inclusion. Consequently our i n i t i a l interval of H u F reduces t o an i n i t i a l interval of H , thus of B . 0 6.5. To end t h i s section, l e t us prove t h a t the axiom of dependent choice, used in 6.2, i s avoidable in view of obtaining GALVIN's theorem.

Come back t o our hypotheses in 6.1. Given a f i n i t e s e t H and a n i n f i n i t e set X > H : e i t h e r (1) 3 i n f x A z' i n f ( Z C y 3~ f i n (FcZ A H u F has $-i.i.))

0

ys

or ( 2 )

3,

inf Y c _ X

A

vz inf(ZcY=> 'dF f i n ( F c Z =>

HuF

has n 0 F i . i . ) ) .

Either ( 2 ) i s f a l s e ; in other words:

vy inf

Y c _ X =>

2 z inf

(ZSY

3, f i n

(FcZAHUF

has 5 i . i . ) ) .

I n such a case, take Y = X ; then take any Y c X b u t change the notation, writing Z instead of Y : we get the following 3, inf Y = X A \dz inf ( Z S X => 3, f i n (FcZ A HuF has F i . i . ) )

So we obtain (1) strengthened by the unambiguous definition

Y = X

.

.

Or ( 2 ) i s t r u e , with (1) true or f a l s e , which i s immaterial. In t h i s case, take a l l the i n f i n i t e s e t s Y which s a t i s f y ( 2 ) , and note t h a t each i n f i n i t e subset of a Y i s s t i l l a Y satisfying ( 2 ) . Let us proceed lexicographically: we take the l e a s t integer uo f o r which there e x i s t s a Y beginning with uo . Then the l e a s t u l , uo f o r which there e x i s t s a Y beginning with uo, u1 ; and so on. Finally we adopt the unambiguous definition Y = i u o , u l , . . . ) .0 EXERCISE 1 - SCHUR NUMBERS 1 - Given a p a r t i t i o n of the s t r i c t l y positive integers into a f i n i t e number of classes called columns, show t h a t a t l e a s t one of the columns contains three d i s t i n c t integers a,b,c with c = a+b (SCHUR 1916). Hint. To each column U associate the class of pairs of integers x , y such t h a t the absolute value I x-yl belongs t o U ; then apply RAMSEY's theorem. 2 - A s e t U of integers i s said t o be additively f r e e i f f the sum of any two integers belonging t o U does not belong t o U Given an integer k , show that there e x i s t s an integer k+, k such t h a t , f o r each p a r t i t i o n of the integers 1,2, ...,k+ i n t o k classes called columns, there is a t l e a s t one nonadditively f r e e column. The smallest k+ will be denoted by s ( k ) and called the Schur number r e l a t i v e t o k Show t h a t s ( 1 ) = 2 , s ( 2 ) = 5 , s(3) = 14 ( s t a r t with the column 5,6,7,8,9).

.

.

Chapter 3

3

-

87

I n 1961, Leonard D. BAUMERT (see p u b l i c a t i o n 1965) e s t a b l i s h e d , w i t h t h e a i d

o f a computer, t h a t s ( 4 ) = 45. Here i s t h e example he gave o f a p a r t i t i o n o f t h e f i r s t 44 s t r i c t l y p o s i t i v e i n t e g e r s i n t o 4 a d d i t i v e l y f r e e columns: 1 2 4 9 3 7 6 1 0 11 5 8 13 15 18 20 12 17 21 22 14 19 24 23 16 26 27 25 29 28 33 30 31 40 37 32 34 42 38 39 35 44 43 41 36 2 Show t h a t t h e Ramsey number ( 3 ) k >/ s ( k ) + l : a s s o c i a t e t o each column U t h e s e t o f p a i r s o f integers from

-

Show t h a t

s(k+l)

gers

1 to

p = s(k)-1

gers

p+l

4

2p + 1

to

.

>/

ned

& 158 , thus

I n general

s(5) 2 (3)5

k

3 ( 3 +1)/2

s(k)

k

>/

.

1

46

.

Begin

with the p a r t i t i o n o f the inte-

columns. Add a

(k+l)st

column o f t h e i n t e -

Then complete each column formed o f i n t e g e r s

Hence

2p+l+u

s ( k ) f o r which t h e a b s o l u t e v a l u e o f t h e

2 (3)4

3.s(k) into

.

integers s(5)

1 to

U ; thus

d i f f e r e n c e belongs t o

+ 159

u by t h e

>, 135 . FREDRICKSON 1979

3 134 and (3);

obtai-

. 3 (3k-4.89/2) + 1/2 f o r k >/ 4 . T h i s

and even

i n e q u a l i t y i s improved by ABBOTT, HANSON 1972 who o b t a i n , i f one r e c t i f i e s t h e i r numerical e r r o r ,

-

EXERCISE 2

1

-

Let

s(k)

3

89(k-7)/4.1201

+

1 for

k

>/

4

.

RAMSEY'S THEOREM WITH m-TUPLES

m, k

be two i n t e g e r s ; t a k e an

m-sequence o f i n t e g e r s

ai (i= 1, ..., m).

P a t t e r n e d a f t e r t h e f i n i t e Ramsey theorem, p r o v e t h a t t h e r e e x i s t s an of i n t e g e r s Fi Xm)

bi

+ ai

o f cardinality

where

subsets

xi

, for

belongs t o Fi

Gi

bi

s a t i s f y i n g t h e f o l l o w i n g . Given an

with -

Gi

i n the Cartesian product

m = 2 with

(kp).kkp

and

p

x

... x

k

Gm

m-tuples

(x

,.... ,

c o l o r s , t h e r e e x i s t s a sequence o f

, such t h a t a l l t h e m-tuples

have t h e same c o l o r . Note t h a t n o t h i n g

p a i r w i s e d i s j o i n t , which i s o f t e n convenient.

Fi al = a2 = p

. Beginning w i t h

a set

F1

o f cardinality

F2 o f c a r d i n a l i t y (p).kkp , we o b t a i n G2 5 F2 o f c a r d i n a l i t y H1 C, F1 o f c a r d i n a l kp , such t h a t , f o r e v e r y o f H1 , a l l t h e o r d e r e d p a i r s w i t h f i r s t t e r m x and second term

and an " i n t e r m e d i a t e " s e t

element

, into

o f c a r d i n a l i t y ai -

G1 x

i s changed i n assuming t h e Example:

every p a r t i t i o n o f the s e t o f Fi

m-sequence

m-sequence of s e t s

THEORY OF RELATIONS

88

c

an element of G2 have the same c o l o r . F i n a l l y we obtain G1 H1 p , w i t h t h e monochromatic Cartesian product G1 x G2 . 2 - For m = 2 and a l = a 2 = 2 , we can take bl = 3 and b2 = 9 sely. W e search f o r a symmetric s o l u t i o n bl = b2 ; show t h a t bl = i n s u f f i c i e n t . On t h e o t h e r hand, f o r the common value 5 , we always

of cardinal

, o r converb2 = 4 i s have a mono-

.

chromatic Cartesian product with Card G1 = Card G2 = 2 One could assume f i r s t t h a t t h e r e e x i s t s an element x of F1 joined t o a t l e a s t 4 elements of F2 by the same c o l o r , denoted ( + ) . Call t h i s subset of F2 of cardinal 4. Then e i t h e r t h e r e e x i s t s another element of F1 y i e l d i n g t h e c o l o r (+) w i t h two elements of , o r two elements of F1 d i s t i n c t from x y i e l d the opposite c o l o r ( - ) w i t h two elements of F; Next, we assume t h a t each element of F1 y i e l d s a p a r t i t i o n of 2 as opposed t o 3 elements of F2 f o r t h e c o l o r s . Such an element x i s c a l l e d (+)-major i t y o r (-)-majority, according t o whether t h e r e a r e 3 edges ( o r p a i r s ) emanat i n g from x with c o l o r (+), o r 3 w i t h c o l o r ( - ) . Then t h e r e e x i s t 3 elements of F1 of t h e same majority: this y i e l d s our conclusion. 3 - Attempt t o extend t h e statement in s e c t i o n 1 above t o t h e case of i n f i n i t e Note t h a t for m = 2 and al = a? = W the c a r d i n a l s a ( i = 1,. ,ml

Fi

Fi

.

,

.. . i = b2 = ~3 do not hold:

values bl we do not have a simple i n f i n i t a r y theorem analogous t o RAMSEY's theorem. Indeed, taking F1 = F2 = is , t h e s e t of i n t e g e r s , i t s u f f i c e s t o c o l o r (+) t h e ordered p a i r s ( x , , ~ , ) i f f x l \ < x2 . However, with t h e axiom of choice, ERDOS, RADO 1956 proved t h a t one can take bl countable and b2 equipotent w i t h t h e continuum, o r conversely: s e e e x e r c i s e 3 below. EXERCISE 3

-

A PARTITION THEOREM WITH POINTS, i . e . ORDERED PAIRS

Take a denumerable s e t D , a s e t C equipotent w i t h t h e continuum, and t h e Cartesian product D x C whose elements ( x , y ) s h a l l be c a l l e d points with t h e abscissa x in D and the o r d i n a t e y i n C P a r t i t i o n the p o i n t s i n t o two c l a s s e s c a l l e d t h e c o l o r s (+) and ( - ) . We s h a l l show t h a t t h e r e e x i s t s either a denumerable s u b s e t Do fo D subset Co of C equipotent w i t h t h e continuum, w i t h Do x Co e n t i r e l y of c o l o r ( - ) , o r two denumerable s u b s e t s D1 0-f D and C1 C with D1 x C1 e n t i r e l y of c o l o r (+) (uses axiom of choice; ERDOS, RADO 1956 p. 482).

.

-

1 Assume t h a t t h e r e does not e x i s t any (-)-monochromatic s e t which i s t h e Cartesian product of a denumerable s e t w i t h a continuum-equipotent set. Take an

a r b i t r a r y denumerable subset X of D and an a r b i t r a r y s u b s e t Y of C which is continuum-equipotent: by hypothesis X x Y i s never (-)-monochromatic. Call Y o t h e projection onto Y of t h e s e t of points of X x Y w i t h c o l o r (+), and note t h a t , by our hypothesis, Card(Y-Yo) i s s t r i c t l y l e s s than t h e

Chapter 3

89

continuum. Every p a r t i t i o n of a continuum-equipotent s e t i n t o countably many c l a s s e s y i e l d s a t l e a s t one c l a s s which i s continuum-equipotent (ch.1 5 4 . 3 , axiom of c h o i c e ) . T h u s there e x i s t s an element xo i n X y i e l d i n g continuum many points w i t h abscissa x o and o r d i n a t e s i n Y with t h e c o l o r (+). 2 - W i t h the s e t s X, Y a s above, f o r each point y in Y , denote by f ( y ) the s e t of a b s c i s s a s x i n X such t h a t the point ( x , y ) has c o l o r (t). Note t h a t f ( y ) i s not always f i n i t e , f o r otherwise t h e r e would e x i s t a continuumequipotent s e t of o r d i n a t e s y w i t h t h e same f ( y ) , hence the same complement X - f ( y ) and f i n a l l y a (-)-monochromatic s e t which i s t h e Cartesian product of a denumerable and continuum-equipotent s e t . Thus t h e r e exists an element y o of Y y i e l d i n g continuum many p o i n t s w i t h o r d i n a t e yo and a b s c i s s a s i n X , having t h e c o l o r (t). 3 - Beginning w i t h Xo = D and Yo = C , take x1 i n Xo y i e l d i n g continuum many points w i t h a b s c i s s a x1 and c o l o r ( + ) . Denote by Y1 t h e continuumequipotent set of o r d i n a t e s . Take y1 i n Y1 y i e l d i n g a denumerable s e t of points with o r d i n a t e y1 and c o l o r ( + ) . Denote by XI t h e denumerable set of {xl} and Y1 - { y l \ , thus a b s c i s s a s . I t e r a t e t h i s , beginning v i t h X1 obtaining an element x2 of t h e f i r s t s e t , and y2 an element of t h e second, w i t h continuum many points with a b s c i s s a x2 and c o l o r (+), and denumerably many points w i t h o r d i n a t e y2 and c o l o r (+). The f o u r points w i t h a b s c i s s a s x l , x2 and o r d i n a t e s y l , y2 have c o l o r ( + ) . By i t e r a t i o n , obtain a denumerable s e t of a b s c i s s a s x i and a denumerable s e t o f o r d i n a t e s y i y i e l d i n g t h e color (t) f o r a l l p o i n t s ( x i , y . ) ( i , j p o s i t i v e i n t e g e r s ) . J 4 - I t follows t h a t i f the plane, which i s the Cartesian product of two sets both equipotent w i t h t h e continuum, i s p a r t i t i o n e d i n t o two c o l o r s , then e i t h e r t h e r e exists a denumerable " g r i d " of t h e f i r s t c o l o r , or t h e r e e x i s t both a g r i d obtained a s t h e product of denumerably many a b s c i s s a s w i t h continuum many o r d i n a t e s , and a g r i d obtained as t h e product of continuum many abscissas with denumerably many o r d i n a t e s , both of t h e second c o l o r . On t h e o t h e r hand, using SIERPINSKI's counterexample i n 3.1, we can p a r t i t i o n t h e plane i n t o f o u r c o l o r s so t h a t every monochromatic g r i d i s a t most the product of a denumerable s e t of a b s c i s s a s w i t h a denumerable s e t of o r d i n a t e s .

-

90

THEORY OF RELATIONS

Problem. Does there e x i s t a p a r t i t i o n of the plane into two colors, or into three colors, such t h a t every monochromatic grid i s a t most the product o f a denumerable s e t with a denumerable set. EXERCISE 4 - SPERNER'S LEMMA Let E be a s e t with f i n i t e cardinality 2h or 2 h + l . Then every s e t o-f sub. s e t s of E which are mutually incomparable with respect t o inclusion has 2 (even case) ( 2 h + l ) ! / h ! ( h + l ) ! (odd case) cardinality a t most ( 2 h ) ! / ( h ! ) In other words, the largest possible cardinality i s obtained by taking the s e t of a l l h-element subsets. Beginning with a s e t Jf o f subsets of E , none of which i s included in another, by replacing i f necessary each s e t by i t s complement, we can always assume t h a t We shall prove t h a t the smallest cardinality of the elements of 4 i s p,< h i f p < h (even case) or p 4 h (odd case) , then we can injectively substitute for every subset A of minimum cardinality p a superset B of A of cardinality p + l . Indeed, i f t h i s i s possible, then the B will be d i s t i n c t

.

or

.

and of the same cardinal p + l , hence will be mutually incomparable with respect t o inclusion. Moreover, no B can be included in an element of of larger cardinality, since then the p-element s e t A from which B was obtained would i t s e l f be included in t h a t s e t . Now note t h a t SPERNER's lemma e a s i l y follows from the l i n e a r independence lemma 5.1 in the case of E f i n i t e and q = 1 For further developments on t h i s subject, see POUZET, ROSENBERG 1982.

4

.