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14 Erdös-Rado Theorem
14 Erdos-Rado Theorem
In the next few lectures we shall give some applications of indiscernible elements. In order to show that models with infinite sets of indiscernibles exist, we need the following partition theorem of ERDOS and RADO[1956]. The Erdos notation a + where a, p, y are cardinals and 1 5 n < o,is useful here. [XI" is the set of all x c X of power exactly n.
(a)!,
(a):
DEFINITION. c1+ means that whenever 1 x1 = a and the set [X]" is written as a union of y sets, [XJ" = U i E Y C ithere , is a subset Y c X of power I Y [ = such that for some i e y, [Y]"c Ci.Equivalently, given any function f on [X]n into y, there exists a subset Y c X of power IYI = p and an i E y such that for all {yl, . .,y,} E [Y]", f ( { y l , . . .,y.}) = i. Note that the relation a + is preserved under increasing the number on the left and decreasing the numbers on the right. Given a cardinal a, we define a0(a) = OL,and for ordinals 5 > 0,
. (a)!
Zt (a) =
sup (2'<(")).E.g., >,(a) = 2". 5't
THEOREM 20 (THEERDOS-RADO THEOREM) Let c1 be an infinite cardinal and n < o.Then Zn(ff)'
PROOF.
+ (a+):+1.
We argue by induction on n. The result for n 75
=
0 is the ob-
76
ERDOS-RADO THEOREM
ti4
vious pigeon hole principle: 'If a set of power x + is divided into a parts, at least one part has power a". (a+):. Let X be a set of power Let n > 0 and assume > n - l ( a ) + >,,(a)+ and let
[ X I " = (J ci i d
where I has power a. We may suppose that I c X and that Ci n Cj if i # j . Let R be the n 2 placed relation on X defined by
+ R(x,... x , , i ) i f f i ~ I a n d { x ,,..., x,,}€Ci.
=
0
Form the model
2l
=
( X , R, i)ier.
Let us say that an elementary submodel B < M is P-saturated relatiue to M iff for every Z c B of power IZI < P, every type C(u) which is realized in (2, b)b& is realized in (23, b),,, . We claim that PI has an elementary submodel 23 of power >,,(a) such that I c B and 2' 3 is >,,-,(a)+-saturated relative to 2. The proof is like the proof of the existence of p+-saturated models of power 28. One need only form an elementary chain
By < % y >,,-I(&)+ such that I c Bo , each Byhas power an(a), and for all Z
c B, of power an-l(a), every type ~ ( u realized ) in (2,b)b,Z is realized in b)b&* This is possible because there are at most >,,(a)'n-'(a)* 2'"-L(u)= >,,(a) such sets Z and types C(u) at stage y.
Since M has power > >,,(a), 2' 3 is a proper submodel of 2. Choose an element c E A - B. We then form a sequence b y 7
Y < an-I(a)+
of elements of B such that for all y, b, realizes the same type as c in (2,b s ) d < f .All the by's are dictinct because c # B. Let U be the set of all by's. Then U has power >n-l(a)f. Partition [U]n into disjoint sets D i , i E I , as follows: for yo < . . . < yn-l < a,,-l(a)+, { by , , . . . . , b y n - , } e D i iff { b , o , . . . , b Y n - l , ~ } E C i .
141
ERDOS-RADO THEOREM
77
By induction hypothesis for n-1, there is a subset Y c U of power IYI > u such that for some i E I,
[Y ] " c Di .
Then for any yo <
. . . < Y , , - ~ < y,,, (by0
,...' b y n - l , c ) ~ C j .
Since by"realizes the same type as c in (H, bs)d
This shows that
[Y]""
c
ci. -1
The above proof is due to S . Simpson, and is simpler than the original purely set-theoretic proof of Erdos and Rado.
In the next few lectures we shall give some applications of indiscernible elements. In order to show that models with infinite sets of indiscernibles exist, we need the following partition theorem of ERDOS and RADO[1956]. The Erdos notation a + where a, p, y are cardinals and 1 5 n < o,is useful here. [XI" is the set of all x c X of power exactly n.
(a)!,
(a):
DEFINITION. c1+ means that whenever 1 x1 = a and the set [X]" is written as a union of y sets, [XJ" = U i E Y C ithere , is a subset Y c X of power I Y [ = such that for some i e y, [Y]"c Ci.Equivalently, given any function f on [X]n into y, there exists a subset Y c X of power IYI = p and an i E y such that for all {yl, . .,y,} E [Y]", f ( { y l , . . .,y.}) = i. Note that the relation a + is preserved under increasing the number on the left and decreasing the numbers on the right. Given a cardinal a, we define a0(a) = OL,and for ordinals 5 > 0,
. (a)!
Zt (a) =
sup (2'<(")).E.g., >,(a) = 2". 5't
THEOREM 20 (THEERDOS-RADO THEOREM) Let c1 be an infinite cardinal and n < o.Then Zn(ff)'
PROOF.
+ (a+):+1.
We argue by induction on n. The result for n 75
=
0 is the ob-
76
ERDOS-RADO THEOREM
ti4
vious pigeon hole principle: 'If a set of power x + is divided into a parts, at least one part has power a". (a+):. Let X be a set of power Let n > 0 and assume > n - l ( a ) + >,,(a)+ and let
[ X I " = (J ci i d
where I has power a. We may suppose that I c X and that Ci n Cj if i # j . Let R be the n 2 placed relation on X defined by
+ R(x,... x , , i ) i f f i ~ I a n d { x ,,..., x,,}€Ci.
=
0
Form the model
2l
=
( X , R, i)ier.
Let us say that an elementary submodel B < M is P-saturated relatiue to M iff for every Z c B of power IZI < P, every type C(u) which is realized in (2, b)b& is realized in (23, b),,, . We claim that PI has an elementary submodel 23 of power >,,(a) such that I c B and 2' 3 is >,,-,(a)+-saturated relative to 2. The proof is like the proof of the existence of p+-saturated models of power 28. One need only form an elementary chain
By < % y >,,-I(&)+ such that I c Bo , each Byhas power an(a), and for all Z
c B, of power an-l(a), every type ~ ( u realized ) in (2,b)b,Z is realized in b)b&* This is possible because there are at most >,,(a)'n-'(a)* 2'"-L(u)= >,,(a) such sets Z and types C(u) at stage y.
Since M has power > >,,(a), 2' 3 is a proper submodel of 2. Choose an element c E A - B. We then form a sequence b y 7
Y < an-I(a)+
of elements of B such that for all y, b, realizes the same type as c in (2,b s ) d < f .All the by's are dictinct because c # B. Let U be the set of all by's. Then U has power >n-l(a)f. Partition [U]n into disjoint sets D i , i E I , as follows: for yo < . . . < yn-l < a,,-l(a)+, { by , , . . . . , b y n - , } e D i iff { b , o , . . . , b Y n - l , ~ } E C i .
141
ERDOS-RADO THEOREM
77
By induction hypothesis for n-1, there is a subset Y c U of power IYI > u such that for some i E I,
[Y ] " c Di .
Then for any yo <
. . . < Y , , - ~ < y,,, (by0
,...' b y n - l , c ) ~ C j .
Since by"realizes the same type as c in (H, bs)d
This shows that
[Y]""
c
ci. -1
The above proof is due to S . Simpson, and is simpler than the original purely set-theoretic proof of Erdos and Rado.