### A 2-coloring of $[1,N]$ can have $(1/22){N}^{2}+O\left(N\right)$ monochromatic Schur triples, but not less.

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A nontrivial surjective Čech closure function is constructed in ZFC.

The paper provides a proof of a combinatorial result which pertains to the characterization of the set of equations which are solvable in the composition monoid of all partial functions on an infinite set.

A dichotomy concerning ideals of countable subsets of some set is introduced and proved compatible with the Continuum Hypothesis. The dichotomy has influence not only on the Suslin Hypothesis or the structure of Hausdorff gaps in the quotient algebra $P\left(\mathbb{N}\right)$/ but also on some higher order statements like for example the existence of Jensen square sequences.

We prove the following conjecture of J. Mycielski: There exists a free nonabelian group of piecewise linear, orientation and area preserving transformations which acts on the punctured disk {(x,y) ∈ ℝ²: 0 < x² + y² < 1} without fixed points.

Assuming large cardinals, we show that every κ-complete filter can be generically extended to a V-ultrafilter with well-founded ultrapower. We then apply this to answer a question of Abe.

We show that splitting of elements of an independent family of infinite regular size will produce a full size independent set.

We prove among other theorems that it is consistent with $ZFC$ that there exists a set $X\subseteq {2}^{\omega}$ which is not meager additive, yet it satisfies the following property: for each ${F}_{\sigma}$ measure zero set $F$, $X+F$ belongs to the intersection ideal $\mathcal{M}\cap \mathcal{N}$.

We present an example of a Banach space $E$ admitting an equivalent weakly uniformly rotund norm and such that there is no $\Phi :E\to {c}_{0}\left(\Gamma \right)$, for any set $\Gamma $, linear, one-to-one and bounded. This answers a problem posed by Fabian, Godefroy, Hájek and Zizler. The space $E$ is actually the dual space ${Y}^{*}$ of a space $Y$ which is a subspace of a WCG space.