A note on a new ideal

TL;DR

This paper introduces a new ideal $\\mathcal{WR}$ and characterizes weak Ramseyness in terms of the Kat\'etov order, revealing its combinatorial properties and implications for ideal convergence and monotonic subsequences.

Contribution

It defines the ideal $\mathcal{WR}$, proves its critical role in weak Ramseyness, and explores its combinatorial and convergence properties, answering an open question.

Findings

An ideal is not weakly Ramsey iff it is above $\mathcal{WR}$ in the Kat\'etov order.

$\mathcal{WR}$ is critical for ideal convergence of quasi-continuous functions.

$\mathcal{WR}$ is not 2-Ramsey, but all isomorphic ideals are Mon.

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 Kat\v{e}tov 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 Filip\'ow 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).