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Ideal version of Ramsey's theorem

Rafał Filipów, Nikodem Mrożek, Ireneusz Recław, Piotr Szuca (2011)

Czechoslovak Mathematical Journal

We consider various forms of Ramsey's theorem, the monotone subsequence theorem and the Bolzano-Weierstrass theorem which are connected with ideals of subsets of natural numbers. We characterize ideals with properties considered. We show that, in a sense, Ramsey's theorem, the monotone subsequence theorem and the Bolzano-Weierstrass theorem characterize the same class of ideals. We use our results to show some versions of density Ramsey's theorem (these are similar to generalizations shown in [P....

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