arXiv subjects

Adam Kwela

Publications and source records attributed to Adam Kwela.

39 records · Page 3Linked to original sources

arXiv2015

A note on a new ideal

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 $ω$ isomorphic to $\mathcal{WR}$ is Mon (every sequence of reals contains a monotone subsequence indexed by a $\mathcal{I}$-positive set).

math.CO↗

arXiv2014

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

We give an exact value of the rank of an $\mathcal{F}$-Fubini sum of filters for the case where $\mathcal{F}$ is a Borel filter of rank $1$. We also consider $\mathcal{F}$-limits of filters $\mathcal{F}_i$, which are of the form $\lim_\mathcal{F}\mathcal{F}_i=\left\{A\subset X: \left\{i\in I: A\in\mathcal{F}_i\right\}\in\mathcal{F}\right\}$. We estimate the ranks of such filters; in particular we prove that they can fall to $1$ for $\mathcal{F}$ as well as for $\mathcal{F}_i$ of arbitrarily large ranks. At the end we prove some facts concerning filters of countable type and their ranks.

math.LO↗

arXiv2013

Topological representations

This paper studies the combinatorics of ideals which recently appeared in ergodicity results for analytic equivalence relations. The ideals have the following topological representation. There is a separable metrizable space $X$, a $σ$-ideal $I$ on $X$ and a dense countable subset $D$ of $X$ such that the ideal consists of those subsets of $D$ whose closure belongs to $I$. It turns out that this definition is indepedent of the choice of $D$. We show that an ideal is of this form if and only if it is dense and countably separated. The latter is a variation of a notion introduced by Todor\vcević for gaps. As a corollary, we get that this class is invariant under the Rudin--Blass equivalence. This also implies that the space $X$ can be always chosen to be compact so that $I$ is a $σ$-ideal of compact sets. We compute the possible descriptive complexities of such ideals and conclude that all analytic equivalence relations induced by such ideals are $\mathbfΠ^0_3$. We also prove that a coanalytic ideal is an intersection of ideals of this form if and only if it is weakly selective.

math.LO↗