Combinatorial Set Theory. by Neil H. Williams

Logic

By Neil H. Williams

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Thus K ' ( K ' , q k ;k < 7)'. 2. Suppose K ' ( K ' , q k ; k < r)', and for a set S with IS1 = K let any partition -+ -+ -+ [S]' = A U U { A k ; k < y} be given. Suppose for each k with k < y that there is no subset H of S with IM = q k and [HI2C Ak. We must show that then there i s H i n [S]" with [HI2 C A . Take a family C = { C;, u < K ' } such that the given partition is canonical in e . We may suppose that always IC,l> K ' . Since q k < K ' < lc,l, for each u we have [C,]' p Ak for each k.

7, one sees IgvI, Ie,l carries over. < 1941 and the rest of the proof CHAPTER 2 ORDINARY PARTITION RELATIONS 8 1. The relations defined The partition relations K + ( q k ; k < 7)" and K + (q); defined below, which are the principal objects of study in this chapter, have now been discussed extensively in the literature. A special case of the symbols first appeared in Erdos and Rado [28], although several results that can be expressed in terms of these symbols had appeared before (see for example Sierpinski [85], Dushnik and Miller [ 111, Erdos [ 131, also Kurepa [64]).

But then [ H U { x A } ] " ' ~ C A k , and IH U { X L } = ~ Vk 1, so the theorem is proved. The results obtained in the last section may now be extended as follows. 2 (GCH). Suppose n 2 1 and y < K ' . 7'hen (i) K(" +) -, , (ii) K ( " + ) + ( K , (~'b)"+' . (KF+' Proof. Of course, when K is regular then (i) follows from (ii). Use induction on n. 7. 5 steps up as follows. 3. Ifn > 1 then (>"(K))+ + (K+);++'. Proof. Again by induction on n. 5. To go from n to n + 1, use the Stepping-up Lemma, noting that ( 2 .

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