We construct a model in which there exists a refining matrix of regular height λ larger than h \documentclass[12pt]{minimal}
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If F is a filter on ω, we say that F is Canjar if the corresponding Mathias forcing does not add
a dominating real. We prove that any Borel Canjar filter is F σ, solving a problem of Hrušák …

We prove that every MAD family can be destroyed by a proper forcing that preserves P-
points. With this result, we prove that it is consistent that ω 1= u< a, solving a nearly 20 year …

Hindman's celebrated Finite Sums Theorem, and its high-dimensional version due to
Milliken and Taylor, are extended from covers of countable sets to covers of arbitrary …

Given a function $ f\in\omega^\omega $, a set $ A\in [\omega]^\omega $ is free for $ f $ if $ f
[A]\cap A $ is finite. For a class of functions $\Gamma\subseteq\omega^{\omega} $, we …

We show that Hechler's forcings for adding a tower and for adding a mad family can be
represented as finite support iterations of Mathias forcings with respect to filters and that …

We construct a model in which there exists a distributivity matrix of regular height $\lambda $
larger than $\mathfrak {h} $; both $\lambda=\mathfrak {c} $ and $\lambda<\mathfrak {c} $ are …

We continue with the study of the Katětov order on MAD families. We prove that Katětov
maximal MAD families exist under This improves previously known results from the …

We introduce a class of cardinal invariants s ℐ for ideals ℐ on ω which arise naturally from
the FinBW property introduced by Filipów et al.(2007). Let ℐ be an ideal on ω. Define s I …

In this survey paper we collect several known results on destroying tall ideals on countable
sets and maximal almost disjoint families with forcing. In most cases we provide streamlined …