A note on a new ideal

In this paper we study a new ideal WRWR. The main result is the following: an ideal is not weakly Ramsey if and only if it is above WRWR 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 WRWR is critical for ideal convergence of sequences of quasi-continuous functions. We study further combinatorial properties of WRWR and weak Ramseyness. Answering a question of Filipów et al. we show that WRWR is not 2-Ramsey, but every ideal on ω   isomorphic to WRWR is Mon (every sequence of reals contains a monotone subsequence indexed by an II-positive set).