A note on a new ideal

Abstract

In this paper we study a new ideal $\mathcal{WR}$. The main result is the following: an ideal is not weakly Ramsey if and only if it is above $\mathcal{WR}$ in the Katětov order. Weak Ramseyness was introduced by Laflamme in order to characterize winning strategies in a certain game. We apply result of Natkaniec and Szuca to conclude that $\mathcal{WR}$ is critical for ideal convergence of sequences of quasi-continuous functions. We study further combinatorial properties of $\mathcal{WR}$ and weak Ramseyness. Answering a question of Filipów et al. we show that $\mathcal{WR}$ is not $2$-Ramsey, but every ideal on $\omega$ isomorphic to $\mathcal{WR}$ is Mon (every sequence of reals contains a monotone subsequence indexed by a $\mathcal{I}$-positive set).

References in corpus (1)

• Ranks of $\mathcal{F}$-limits of filter sequences

Cited by in corpus (5)

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• Erdős-Ulam ideals vs. simple density ideals
• Ideal weak QN-spaces
• Ideal equal Baire classes
• On the structure of Borel ideals in-between the ideals $\mathcal{E}\mathcal{D}$ and $\mathrm{Fin}\otimes \mathrm{Fin}$ in the Katětov order