On the height of order ideals

Gábor Czédli; Miklós Maróti

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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...