Incomparability and Intersection Properties of Boolean Interval Lattices and Chain Posets

Europ. J. Combinatorics (1996) 17 , 677 – 687

Incomparability and Intersection Properties of Boolean Interval Lattices and Chain Posets RUDOLF AHLSWEDE

NING CAI

AND

In a canonical way, we establish an AZ-identity (see [2]) and its consequences, the LYM-inequality and the Sperner property, for the Boolean interval lattice. Furthermore, the Bollobas inequality for the Boolean interval lattice turns out to be just the LYM-inequality for the Boolean lattice. We also present an Intersection Theorem for this lattice. Perhaps more surprising is that by our approach the conjecture of P. L. Erdo¨ s et al. [7] and Z. Fu¨ redi concerning an Erdo¨ s – Ko – Rado-type intersection property for the poset of Boolean chains could also be established. In fact, we give two seemingly elegant proofs. ÷ 1996 Academic Press Limited

1. THE BOOLEAN INTERNAL LATTICE (n The main objects of our investigation are collections of intervals in the Boolean lattice @n ; that is, the family of all subsets of [n ] 5 h1 , 2 , . . . , n j endowed with union and intersection as lattice operations. For A , B P @n , we define the interval [A , B ] 5 hC : A ’ C ’ B j

(1.1)

in @n , and if A ’ / B here, then we speak of the empty interval I[ . The Boolean interval lattice (n is the set of all intervals in @n endowed with the following ‘meet’ and ‘join’ operations denoted by ‘ ∧ ’ and ‘ ∨ ’: [A , B ] ∧ [A9 , B 9] 5 hC : C P [A , B ] > [A9 , B 9]j 5

HI[A,< A9, B > B9], [

if A < A9 ’ B > B 9 , otherwise,

[A , B ] ∨ [A9 , B 9] 5 [A > A9 , B < B 9].

(1.2) (1.3)

The lattice properties are readily verified. Note that the meet can be viewed as a Boolean intersection. However, for the join we have [A , B ] ∨ [A9 , B 9] ” [A , B ] < [A9 , B 9] and often there is no equality. Clearly, we can define a partial order ‘ < ’ by [A , B ] < [A9 , B 9] ï [A , B ] ’ [A9 , B 9]

or (equivalently)

A9 ’ A ’ B ’ B 9.

(1.4)

We define a rank function r : (n 5 N < h0j by

r ([A , B ]) 5

H0B, \ A 1 1, u

u

if [A , B ] 5 I[ , if [A , B ] ? I[ .

One readily verifies that r is upper semimodular; that is,

r ([A , B ] ∨ [A9 , B 9]) 1 r ([A , B ] ∧ [A9 , B 9]) < r ([A , B ]) 1 r ([A9 , B 9]). This is not used in this paper. However, we frequently use an equivalent description of non-empty intervals. 677 0195-6698 / 96 / 080677 1 11 $18.00 / 0

÷ 1996 Academic Press Limited

678

R. Ahlswede and N. Cai

Instead of [A , B ] we write kC , D l , where C 5 A and D 5 B \ A. Note that C > D 5 [ and, given any disjoint subsets C and D of [n ] , kC , D l is the interval corresponding to [C , C < D ]. Now kC , D l < kC 9 , D 9l ï C 9 ’ C and (C \ C 9) < D ’ D 9 é C 9 ’ C and D ’ D 9. One also readily verifies that kC , D l ∧ kC 9 , D 9l 5 kC < C 9 , D > D 9l ,

if kC , D l ∧ kC 9 , D 9l ? I[ .

(1.5)

Finally (with a little abuse of notation), we also use r for the second interval description: r (kC , D l) 5 uD u 1 1. (1.6) 2. THE AZ-IDENTITY,

THE

LYM-INEQUALITY,

AND THE

SPENCER PROPERTY

FOR

(n

Let us introduce @nk 5 ( [nk] ) and let us denote by ( kn the set of intervals from (n of rank k (0 < k < n 1 1). Observe first that, for all I P ( kn , uhI 9 P ( kn11: I 9 ” I ju 5 n 2 k 1 1

(2.1)

uhI 9 P ( kn21: I 9 ’ I ju 5 2(k 2 1).

(2.2)

and that This regularity property of a lattice is sufficient for the LYM-inequality to hold. We move directly to the AZ-identity. For any ! ’ (n and any I 5 [A , B ] P (n with

!I 5 hK P ! : K ’ I j ? [

(2.3)

!I 5 h[Ai , Bi ]: 1 < i < a j

(2.4)

write and define

S U

a

W! (I ) 5 uB u 2 ! Ai i 51

UD 1 SU " B U 2 A D . a

i 51

i

u u

(2.5)

If (2.3) does not hold, set W! (I ) 5 0 . THEOREM 1 (AZ-identity). For any ! ’ (n ,

O

I P(n

W! (I )

S

2n2r (I )12(r (I ) 2 1)

n r (I ) 2 1

D

; 1.

PROOF. By (2.1) (or (2.2)), (n has exactly

P (n 2 k 1 1)Sor 2 P (k 2 1)D 5 2 n!

n 11

2n

n 11

n

k 51

n

k 52

maximal chains. Also, exactly (n 2 r (I ) 1 1)! W! (I )2r (I)22(r (I ) 2 2)! maximal chains leave the upset 8 (! ) 5 hK P (n : 'I 9 P ! with K > I 9j in I 5 [A , B ]. Since r (I ) 5 uB \ Au 1 1 , we obtain

O

I P8(! )

(n 2 r (I ) 1 1)! W! (I )2r (I )22(r (I ) 2 2)! 5 2nn !

Since W! (I ) 5 0 for I ¸ 8(! ) , the identity follows.

h

Properties of Boolean lattices and posets

679

The Whitney numbers wk of (n are defined by wk 5 u( knu

for 0 < k < n 1 1 .

(2.6)

They can be evaluated. LEMMA 1. We hay e : n n 2k 11 (i) wk 5 ( k 2 for 0 , k < n 1 1 and w0 5 1 and , consequently , 1 )2 n 11 (ii) u(n u 5 ok50 wk 5 3n 1 1 . PROOF. The set ( kn is exactly the set of intervals kC , D l with D P @kn 21 and 1 k C ’ D c 5 [n ] \ D. Therefore (i) holds and (ii) follows, because u(n u 5 onk1 h 50 u( nu . COROLLARY 1 (LYM-inequality). For any antichain ! ’ (n ,

O ! w> (

n 11

u

k 50

k nu

< 1.

k

PROOF. For ( P ! , by the antichain property and by (2.5), W! (I ) 5 (uB u 2 uAu) 1 (uB u 2 uAu) 5 2(r (I ) 2 1) and, by Theorem 1,

O

2(r (I ) 2 1)

I P!

S

2n2r (I )12(r (I ) 2 1)

Using Lemma 1(i), we obtain

O w1

I P!

r (I )

n r (I ) 2 1

D

< 1.

<1 h

and thus the result. COROLLARY 2 (Sperner property). (i) For ey ery antichain ! ’ (n , u! u <

max wk 5

0,k
(ii) max wk 5

0,k
5

A

B

n n 1 1 11 3 n 11 2n2 – . 21 3

wl11 , wl11 , wl11 5 wl ,



if n 1 1 5 3l 1 1 , if n 1 1 5 3l 1 2 , if n 1 1 5 3l.

(iii) The antichains ! of maximal length are

!5

H(( , , l 11 n l n

if n 1 1 5 3l 1 m , if n 1 1 5 3l.

m 5 0, 1, 2,

Thus , if 3 u n 1 1 , then there are two optimal antichains. PROOF. Corollary 1 implies that u! u < max0,k 11 max(wk21 , wk11) gives the necessary condition for wk to be maximal:  n– 3 
680

R. Ahlswede and N. Cai

Let ! be a antichain of maximal length wl . By Theorem 1 (or even Corollary 1), necessarily, u! > ( lnu 1 u! > ( ln11u 5 wl and, again by Theorem 1, W! (I ) 5 0 for I ¸ !. Now suppose that ! ? ( ln , ( ln11. For all I 5 [A , B ] P ( ln11 \ ! , by its definition W! (I ) 5 0 implies that, for all x P B \ A , A9 5 A < hx j satisfies [A9 , B ] P ! > ( nl and B 9 5 B \ hx j satisfies [A , B 9] P ! > ( ln. This means that all sub-intervals of I , which have a rank l , are in ! > ( ln. However, no I P ! > ( ln11 has a sub-interval in ! > ( ln. This means that the bipartite graph (( ln , ( ln11 , ’ ) has two connected components. This is impossible, because I 5 [A , B ] P ( ln11 is connected to J 5 [[ , B 9] P ( ln11 by alternating deleting elements from A and B and each two vertices [f , B 9] and [f , B 0] are connected in this graph. h 3. INTERSECTING SYSTEMS

IN

(n

AND

AND

( kn ,

THE

ERDO¨ S – KO – RADO PROPERTY

UNIQUENESS

The goal of our investigations is to understand how intersecting systems, which have been studied extensively in Boolean lattices [see [4, 6]) and also in other structures (see [4] and [5, 8 – 10, 14, 15]), behave in (n and ( kn. We call S ’ (n an intersecting system , if for all I , I 9 P S , I ∧ I 9 5 I > I 9 ? [.

(3.1)

Also, we say that S is saturated , if it is not a proper subset of an intersecting system. A simple and basic saturated intersecting system is, for C ’ [n ] ,

(n (C ) 5 hI P (n : C P I j.

(3.2)

We show first that its cardinality is independent of C. LEMMA 2.

For all C ’ [n ] ,

u(n (C )u 5 2n.

PROOF. The intervals containing C are of the form [A , B ] , A ’ C ’ B. Clearly, there are 2uCu2n 2uC u such intervals. h Next we show that all saturated intersecting systems are of the form (3.2). LEMMA 3. For ey ery intersecting system S ’ (n a D P (n , D ? I[ , exists with D < I for all I P S. Furthermore , if S is saturated , then S 5 (n (C ) for some non -empty C ’ [n ]. PROOF. Write S 5 h[At , Bt ]: t P T j and note that Ai ‘ Bj (i , j P T ) implies ! At ’ " Bt .

t PT

t PT

(3.3)

So, the interval D 5 [!t PT At , "t PT Bt ] satisfies D ? I[ and D < I (I P S ). Furthermore, when S is saturated, then D 5 [C , C ] for some C ’ [n ] and D P S , S 5 (n (C ). h Whereas in (n the intersecting systems (n (C ) ([ ? C ’ [n ]) are exactly the largest intersecting systems, in @n the systems hX ’ [n ]: x P X j (x P [n ]) are not the only intersecting systems of maximal cardinality. However, connections between these lattices can be established via their Whitney numbers. For this, we define ( kn(C ) 5 (n (C ) > ( kn. (3.4)

Properties of Boolean lattices and posets

681

n LEMMA 4. u( kn(C )u 5 ( k 2 1 ).

PROOF. Note that

H

( kn(C ) 5 kC > E c , E l: E P

Sk[2n]1DJ 5 H[C \ E, C < E]: E P Hk[2n]1DJ h

and thus the claimed identity.

We now consider intersecting systems of intervals of rank k. This is analogous to the case of k elements sets, considered originally in [6]. It is remarkable that in the new situation we have uniqueness in the sense that only the ( kn(C )’s appear as optimal systems. THEOREM 2.

For ey ery intersecting system S ’ ( kn , S<

Sk 2n 1D

and the ( kn(C ) (C ’ [n ] , 1 < uC u < n 2 k 1 1) are exactly the intersecting systems achiey ing equality. The analysis proceeds in terms of a useful concept of parallelism. We say that the interval kC , D l P (n \ hI[j has direction d (kC , D l) 5 D. The empty interval I[ has no direction. Intervals with the same direction are called parallel. We write I i I 9 , if I and I 9 are parallel. Obviously,

r (I ) 5 r (I 9) ,

if I i I 9.

(3.5)

The next property is familiar from geometry. LEMMA 5.

Parallel intery als are disjoint or , formally , I i I 9 , I ? I 9 é I > I 9 5 I ∧ I 9 5 I[ .

PROOF. For I 5 kC , D l 5 [C , C < D ] , I 9 5 kC 9 , D l 5 [C 9 , C 9 < D ] , C ? C 9 and C > D 5 C 9 > D 5 [ we have I > I 9 5 [C < C 9 , (C < D ) > (C 9 < D )] 5 [C < C 9 , (C > C 9) < D ] 5 I[ , because (C < C 9) > D 5 [ and C < C 9 ’ / C > C 9 for C ? C 9. Consequently, C < C9 ‘ / (C > C 9) < D. h Using this result one readily verifies the next statements, which shed new light on Lemmas 1 and 4. LEMMA 6. (i) For ey ery direction D ’ [n ] , the intery als kC , D l and C ’ D c partition (n . (ii) (n can be partitioned into 2n families of parallel intery als. n n 2k 11 (iii) ( kn can be partitioned into ( k 2 parallel intery als. 1 ) families , each hay ing 2

682

R. Ahlswede and N. Cai

n PROOF OF THEOREM 2. Clearly, Lemma 6(iii) and Lemma 5 imply that uS u < ( k 2 1 ).

Furthermore, by Lemma 3, S ’ (n (C ) , and since by assumption S ’ ( kn , we conclude that S ’ (n (C ) > ( kn. If S is optimal, then necessarily S 5 (n (C ) > ( kn 5 ( kn(C ) , by Lemma 3. h REMARK 1. It is natural also to consider intersecting systems with a qualified constraint. We call S ’ (n d -intersecting if, for all I , I 9 P S ,

r (I ∧ I 9) > d.

(3.6)

Our previous definition is included in the case d 5 1 . There is a simple reduction for the cases d > 2 . For I 5 kC , D l , I 9 5 kC 9 , D 9l , (3.6) is equivalent to kC , D l ∧ kC 9 , D 9l 5 kC < C 9 , D > D 9l with uD > D 9u > d 2 1 and D ? D 9. Therefore for every d -intersecting system S ’ (n there corresponds a (d 2 1)intersecting system of @n $ 5 hD : 'kC , D l P S j of the same cardinality. Conversely, to every (d 2 1)-intersecting system of @n $ there corresponds a d -intersecting system of (n of the same cardinality; namely, for any E P @n , S 5 hkE \ D , D l: D P $ j. Similarly, there is such a correspondence between intersecting systems S ’ ( kn and $ ’ @kn21. 4. FROM LOCAL

TO

GLOBAL INTERSECTION

OF

INTERVALS

AND

INTERSECTING ANTICHAINS

The fact that parallel intervals are disjoint (Lemma 5) has a useful extension. LEMMA 7. necessarily

For two non -disjoint intery als kC1 , D1l and kC2 , D2l , with D1 ’ D2 , kC 1 , D1l < kC2 , D2l

or (equiy alently ) [C 1 , C 1 < D1] ’ [C 2 , C 2 < D2]. PROOF. Recall the definition (1.4) of the partial order. By our assumption, for some X ’ [n ] , C 1 ’ X ’ C 1 < D1 , C 2 ’ X ’ C 2 < D2 and therefore C 1 ’ C 2 < D2 , (4.1) C 2 ’ C 1 < D1 . Since D1 ’ D2 , we conclude first that C 1 < D1 ’ C 2 < D2 . Since C2 > D1 ’ C 2 > D2 5 [ and C 2 ’ C 1 < D1 we conclude further that C 2 ’ C1 .

(4.2) (4.3)

Finally, (4.2) and (4.3) say that C 2 ’ C 1 ’ C 1 < D1 ’ C 2 < D2 and thus [C1 , C1 < D1] ’ [C 2 , C 2 < D2].

h

Properties of Boolean lattices and posets

683

A well-known inequality of Bollobas [3] states that, for any intersecting antichain ^ ’ @n ,

O ^n>2B1 < 1. Sk 2 1 D

n /2

u

k nu

(4.4)

k 51

What is the Bollobas-type inequality for (n ? The answer follows by simple reasoning. For an intersecting antichain 6 5 hkCi , Di l: 1 < i < m j in (n by Lemma 7 necessarily hDi : 1 < i < m j is an antichain in @n . We obtain the following inequality. THEOREM 3.

For an intersecting antichain 6 in (n ,

O 6 >n ( < 1. Sk 2 1 D

n 11

u

k nu

k 51

Conversely, we can translate this inequality backwards. Thus the LYM-inequality for the Boolean lattice is exactly the Bollobas-type inequality for the Boolean interval lattice.

5. AN INTERSECTION THEOREM

FOR

CHAIN POSETS

We now introduce chain posets and prove for them an intersection property conjectured by Erdo¨ s et al. in [7]. There and also by Fu¨ redi (according to [7]), this conjecture has been verified in over large range of parameters. The methods used do not seem to be suitable to settle the conjecture. Our approach does this, and is very simple. In Section 6 we give an even simpler and more direct proof. A strictly increasing sequence of subsets of [n ] and of length k is claled a k -chain. # kn 11 k denotes the set of all those chains and we define #n 5 !nk5 1 # n. The chain C 5 hC 1 ’ C2 ’ ? ? ? ’ Cl j is contained in the chain C9 5 hC 91 ’ C 92 ’ ? ? ? ’ C 9l9j , if hCn : 1 < i < l j ’ hC 9i : 1 < i < l 9j. We denote this containment by ‘ ’ ’. Then (#n , ’) is a poset, which we call the poset of chains (on an n -set). With the chain C we associate an interval conv(C) 5 [C 1 , Cl ] P (n and, conversely, with an interval I P (n we associate the set of chains

#n (I ) 5 hC P #n : conv(C) 5 I j.

(5.1)

Furthermore, for any set of chains # ’ #n we consider the subset of chains

# (I ) 5 hC P # : conv(C) 5 I j.

(5.2)

Similarly # kn(I ) are the k -chains with convex hull I. For fixed k and n , u# kn(I )u depends only on r (I ) 5 r , say, and shall be denoted by q (r ). Clearly, q (r ) 5 0 , if r , k. Now we consider intersecting chains. Two chains C and C9 are intersecting if, for l some pair (i , i 9) , Ci 5 C 9i9. We write C ,C9. A family of chains # is intersecting if l

C , C9 for all C, C9 P # . The maximal cardinality of such a family shall be M (n ). If only k -chains are permitted in # , then we denote the maximal cardinality of u# u by M (n , k ).

684

R. Ahlswede and N. Cai

We say that # is a simple intersecting family if, for some X ’ [n ] , all chains in # have X as a member (or meet X ). CONJECTURE ESS & F. M (n , k ) is assumed by the simple intersecting family all k -chains meeting [ (or [n ]). Since [

# kn 5

!

n] I5[f ,B ] ,B P( r[2 1 ) ,r >k

and, therefore, u[ # knu 5

# kn(I )

O q(r)Sr 2n 1D,

[

# kn of

(5.3)

n 11

(5.4)

r 5k

the conjecture can be restated as M (n , k ) 5

O q(r)Sr 2n 1D.

n 11

(5.5)

r 5k

THEOREM 4. The intersecting family of all k -chains in @n starting with the empty set [ has the maximal cardinality M (n , k ). PROOF. Let # be an intersecting family of k -chains of cardinality u# u 5 M (n , k ). Introduce (n (# ) 5 hconv(C): C P # j and observe that (recalling (5.2)) (# (I ))I P(n(# ) is a partition of # . Now write (n (# ) 5 h[Ai , Bi ]: i P T j. Note also that the intersection property of the chains implies that (n (# ) is an intersecting system of intervals. Therefore Ai ’ Bj for all i , j P T and hence, for all i P T , Ai ’ " Bj 5 B I (say) ’ Bi , jPT

i.e. B I PI

for all I P (n (# ).

This means that, in the terminology of Section 3, n 11

(n (# ) ’ (n (B I ) 5 ! ( rn(B I) r 50

and M (n , k ) 5 u# u 5 <

O O

I P(n (# )

I P(n (B I)

O q(r (I)) n q (r (I )) 5 O O q (r ) 5 O S Dq(r) r 21

u# (I )u <

I P(n (# ) n 11

n 11

r 50 I P( rn(B I)

r 50

(by Lemma 4). h

REMARK 2. q (r ) equals the number of ways in which a set of r 2 1 elements can be partitioned into a sequence of k 2 1 non-empty subsets. This observation gave us the idea of constructing the more direct proof of Theorem 4 in Section 6.

Properties of Boolean lattices and posets

6. A DIRECT PROOF

OF

685

THEOREM 4

With the chain A 5 hA1 ’ A2 ’ ? ? ? > Ak j P # , we associate the sequence of disjoint non-empty sets k n

C 5 A1 ,

D1 5 A2 \ A1 ,

D2 5 A3 \ A2 ,

Dk21 5 Ak \ Ak 21 .

...,

(6.1)

Conversely, from kC , D1 , . . . , Dk21l we can recover A via the equations A1 5 C ,

S

j 21

Aj 5 C < ! Di i 51

D

for j > 2.

(6.2)

Thus we have an alternate representation of chains: kC , D1 , . . . , Dk21l. We now study the intersection property of chains in this terminology. Recall that parallel intervals are disjoint (Lemma 5 in Section 3). l Therefore, for intervals kC 9; D1l , kC ; D1l implies that C 5 C 9 and thus kC 9; D1l 5 kC ; D l. This fact generalizes. LEMMA 8. l

kC 9; D1 , . . . , Dk21l , kC ; D1 , . . . , Dk21l é C 5 C 9 and kC 9; D1 , . . . , Dk21l 5 kC ; D1 , . . . , Dk21l. PROOF. There are j and j 9 with

S

D

i 21

S

j 921

D

C < ! Di 5 C 9 < ! Di . i 51

i 51

(6.3)

We subtract C < C 9 from both sides and obtain, using (6.1), j 21

S S DD Ñ (C < C9) 5 SC 9 < S ! D DD Ñ (C < C 9) 5 ! D . j21

! Di 5 C < ! Di

i 51

i 51

j 921 i 51

Now, necessarily, C 5 C 9.

j921

i

i 51

i

h

PROOF OF THEOREM 4. By Lemma 8, for a family # of intersecting k -chains u# u does not exceed the cardinality d (n , k ) of the set $ kn of all distinct sequences (D1 , . . . , Dk21) with Di > Dj 5 [ (i ? j ) and Di ’ [n ] for i 5 1 , . . . , k 2 1 . Thus M (n , k ) < d (n , k ). Instead of determining d (n , k ) by counting we just observe that there is a bijection  : $ kn 5 hk[ , D1 , . . . , Dk21l: (D1 , . . . , Dk21) P $ knj. The image is exactly [ # kn and its optimality is thus proved. REMARK 3. Inspection of the proofs shows that the condition Di ? [ (i 5 1 , . . . , k 2 1) was not used. Therefore also the intersection problem for chains of length
686

R. Ahlswede and N. Cai

Applying a shifting operator as in [7], one can show that there is an optimal family ^ with a strong d -intersection property saying that for any C, C9 P ^ there is a subset S ’ h0 , 1 , . . . , n j , uS u > d , such that all Xj 5 h1 , 2 , . . . , j j with j P S are contained in both, C and C9. Here X 0 5 [. This means that there is a set 6 5 hS (F ): F P ^ j of subsets of h0, 1, . . . , n j with uS (F ) > S (F 9)u > d

associated with F , such that for j P S (F ) , Xj is contained in F . It seems natural to conjecture that for some » . 0 and n large for d < n (1 2 » ) there is an optimal family of d -intersecting chains all containing X 0 , X 1 , . . . , Xd 21 . The restriction on d is essential, because otherwise the guess is false. To see this, let us assume that n 2 d is bounded by a constant b. Then the number of chains in the family just specified is bounded by a function of b only. However, the family of chains containing hXi ji PS , where S runs through all subsets of h0, 1, . . . , n j with cardinality at least (n 1 d ) / 2 , is d -intersecting and increases with n. Our last contribution is in the spirit of this construction. CONJECTURE. For all n ,d and some w > d there is an optimal d -intersecting family of chains, which contain at least (w 1 d ) / 2 members of X 0 , X 1 , . . . , Xw21 . 7. ANOTHER DESCRIPTION

FOR

(n

Consider 2 (n 5 (n \ hI[j. We express C ’ [n ] as a binary sequence c n 5 (c1 , . . . , cn ) of length n. An interval [A , B ] can thus be described by a pair [a n , b n ] , where at < bt for t 5 1 , . . . , n. This pair in turn can be described by one sequence z n 5 (z1 , . . . , zn ) , with zt 5 2 2 (at 1 bt ) , t 5 1 , 2 , . . . , n. (7.1) We also write z n 5 w ([A , B ]). Conversely, z n determines [A , B ] uniquely. Moreover, if z n 5 w ([A , B ]) and z 9n 5 w ([A9 , B 9]) , then [A , B ] < [A9 , B 9] ï for every t P [n ] with z 9t ? 1 , zt 5 z 9t holds.

(7.2)

Therefore the poset 2 (n can be expressed as a product of posets. [A , B ] P ( kn11 , k > 0 , means that, for z n 5 w ([A , B ]) , uhzt : zt 5 1 , t P [n ]ju 5 k. Two intervals [A , B ] and [A9 , B 9] P 2 ( kn11 are intersecting iff uzt 2 z 9t u ? 2 for all t P [n ] and z n and z 9n do not have the 1’s in exactly the same k positions. This again allows us to characterize the intersecting systems in ( nk11 of maximal cardinality ( nk ). We already have mentioned the system

( kn(C ) 5 hkC \ D , D l: D P @knj ,

C ’ [n ].

(7.3)

Here kC \ D , D l 5 [C \ D , C < D ] and w ([C \ D , C < D ]) 5 z n , with zt 5

5

0 1 2

for t P C \ D , for t P D , for t ¸ C < D.

By Lemma 3, for a maximal intersecting system S , necessarily S 5 hkED , D l: D P @knj.

(7.4)

Properties of Boolean lattices and posets

687

Let Ti (i 5 0 , 2) be the components in which an i occurs for some z n. Then T0 > T2 5 [. Furthermore, T0 < T2 5 [n ] , because otherwise not all D P @kn are used. Define C 5 T0 and observe that ED 5 C \ D. Thus all optimal systems are of the form (7.3). REFERENCES 1. R. Ahlswede and N. Cai, A generalization of the AZ identity, Combinatorica , 13(3) (1993), 241 – 247. 2. R. Ahlswede and Z. Zhang, An identity in combinatorial extremal theory, Ady . Math. , 80(2) (1990), 137 – 151. 3. B. Bolloba´ s, On generalized graphs, Acta Math. Acad. Sci. Hungar. , 16 (1965), 447 – 452. 4. M. Deza and P. Frankl, Erdo¨ s – Ko – Rado theorem—22 years later, SIAM J. Alg. Discr . Methods , 4 (1983), 419 – 431. 5. P. Erdo¨ s and V. T. So´ s, Problems and results on intersections of set systems of structural type, Util. Math. , 29 (1986), 61 – 70. 6. P. Erdo¨ s, C. Ko and R. Rado, Intersection theorems for systems of finite sets, Q. J. Math. Oxford , Ser. 2 , 12 (1961), 313 – 318. 7. P. L. Erdo¨ s, A. Seress and L. A. Sze´ kely, On intersecting chains in Boolean algebras, Preprint, April 1993. 8. R. J. Fandree, R. H. Schelp and V. T. So´ s, Some intersection theorems on two-valued functions, Combinatorica , 6(4) (1986), 327 – 333. 9. P. Frankl and Z. Fu¨ redi, Finite projective spaces and intersecting hypergraphs, Combinatorica , 6(4) (1986), 335 – 354. 10. R. L. Graham, M. Simonivits and V. T. So´ s, A note on the intersection properties of subsets of integers, J. Combin. Theory , Ser. A , 28 (1980), 106 – 116. 11. A. J. W. Hilton, Intersecting and non-union antichains, Preprint, May 1993. 12. D. Lubell, A short proof of Sperner’s theorem, J. Combin. Theory , 1 (1966), 299. 13. L. D. Meshalkin, A generalization of Sperner’s theorem on the number of subsets of a finite set. Theoret. Prob. Applic. , 8 (1963), 203 – 204. 14. M. Simonovits and V. T. So´ s, Intersections on structures, Combinatorial Mathematics , Optimal Design and Applications , Ann. Discr. Math. , 6 (1980), 301 – 314. 15. M. Simonovits and V. T. So´ s, Intersection theorems for subsets of integers, Europ. J. Combin. , 2 (1981), 363 – 372. 16. E. Sperner, Ein Satz u¨ ber Untermengen einer endlichen Menge, Math. Z. , 27 (1928), 544 – 548. 17. K. Yamamoto, Logarithmic order of free distribution lattices, J. Math. Soc. Japan , 6 (1954), 343 – 353. Receiy ed 15 June 1993 and accepted 2 September 1994 RUDOLF AHLSWEDE AND NING CAI Uniy ersita¨ t Bielefeld , Fakulta¨ t fu¨ r Mathematik , Postfach 100 131 , D-4800 Bielefeld 1 , Germany