Combinatorial Set Theory by Lorenz J. Halbeisen

Author:Lorenz J. Halbeisen
Language: eng
Format: epub
Publisher: Springer International Publishing, Cham

(b) For any n ∈ ω, for any positive integer r ∈ ω, and for every colouring π: [ω] n → r, there exists an which is homogeneous for π.

(c) Let {r k : k ∈ ω} and {n k : k ∈ ω} be two (possibly finite) sets of positive integers, and for each k ∈ ω let be a colouring. Then there exists an which is almost homogeneous for each π k .

It is time now to address the problem of the existence of Ramsey ultrafilters. On the one hand, it can be shown that there are models of ZFC in which no Ramsey ultrafilters exist (see Proposition 26.23). Thus, the existence of Ramsey ultrafilters is not provable in ZFC. On the other hand, if we assume, for example, CH (or just ), then we can easily construct a Ramsey ultrafilter.

Proposition 11.9.

If , then there exists a Ramsey ultrafilter.

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