Related papers: Filter-linkedness and its effect on preservation o…

We prove that if two families $\mathcal{F} \subseteq \binom{[n]}{k}$ and $\mathcal{F}' \subseteq \binom{[n]}{k'}$ satisfy $\sum_{1 \leq i, j \leq \ell} \lvert F_i \cap F_j' \rvert \geq \ell^2t - \ell +1$ for every choice of distinct $F_1,…

Based on the well-known cardinal characteristics $\mathfrak{s}$, $\mathfrak{r}$ and $\mathfrak{i}$, we introduce nine related cardinal characteristics by using the notion of asymptotic density to characterise different intersection…

We prove that the Lipshitz-Ozsv\'ath-Thurston correspondence between extended type D structures of knot complements and $\mathbb{F}[U, V]/(UV)$ knot Floer complexes can be arranged so that $\iota_K$-invariant splittings of knot Floer chain…

This paper contributes to the set-theoretic side of understanding Keisler's order. We consider properties of ultrafilters which affect saturation of unstable theories: the lower cofinality $\lcf(\aleph_0, \de)$ of $\aleph_0$ modulo $\de$,…

For a given partially ordered set (poset) and a given family of mappings of the poset into itself, we study the problem of the description of joint fixed points of this family. Well-known Tarski's theorem gives the structure of the set of…

This paper continues the study of the Ramsey-like large cardinals. Ramsey-like cardinals are defined by generalizing the characterization of Ramsey cardinals via the existence of elementary embeddings. Ultrafilters derived from such…

We study the strength of well-founded ultrafilters on ordinals above choiceless large cardinals and their associated Prikry forcings. Gabriel Goldberg showed that all but boundedly many regular cardinals above a rank Berkeley cardinal carry…

Much recent work in cardinal characteristics has focused on generalizing results about $\omega$ to uncountable cardinals by studying analogues of classical cardinal characteristics on the generalized Baire and Cantor spaces $\kappa^\kappa$…

We investigate families of partitions of omega which are related to special coideals, so-called happy families, and give a dual form of Ramsey ultrafilters in terms of partitions. The combinatorial properties of these…

Three families of posets depending on a nonnegative integer parameter $m$ are introduced. The underlying sets of these posets are enumerated by the $m$-Fuss Catalan numbers. Among these, one is a generalization of Stanley lattices and…

We study the problem of approximating the corner polyhedron using intersection cuts derived from families of lattice-free sets in $\mathbb{R}^n$. In particular, we look at the problem of characterizing families that approximate the corner…

Given a family $\F$ of posets closed under disjoint unions and the operation of taking convex subposets, we construct a category $\C_{\F}$ called the \emph{incidence category of $\F$}. This category is "nearly abelian" in the sense that all…

Using the KKS inequalities, we establish bounds on the numbers of $k$-sets in the increasing families generated by self-dual clutters (i.e., clutters $\mathcal{A}$ that coincide with the blockers $\mathfrak{B}(\mathcal{A})$) on their ground…

This article extends a paper of Abraham and Bonnet which generalised the famous Hausdorff characterisation of the class of scattered linear orders. Abraham and Bonnet gave a poset hierarchy that characterised the class of scattered posets…

We investigate which filters on $\omega$ can contain towers, that is, a modulo finite descending sequence without any pseudointersection (in $[\omega]^\omega$). We prove the following results: - Many classical examples of nice tall filters…

We propose a functorial framework for persistent homology based on finite topological spaces and their associated posets. Starting from a finite metric space, we associate a filtration of finite topologies whose structure maps are…

Let $X$ be a compact metric space and let $|A|$ denote the cardinality of a set $A$. We prove that if $f\colon X\to X$ is a homeomorphism and $|X|=\infty$ then for all $\delta>0$ there is $A\subset X$ such that $|A|=4$ and for all $k\in Z$…

We consider the problem of determining the maximum number of pairs $F\subseteq F'$ in a family $\mathcal{F}\subseteq 2^{[n]}$ that avoids certain posets $P$ of height 2. We show that for any such $P$ the number of pairs is…

For each finite poset $F$ with $|F| > 1$, $\chi_{ac}(F)$ denotes the smallest integer $n$ (if it exists) such that the elements of every finite poset $P$ with $|P| > 1$ can be coloured with at most $n$ colours so that every maximal $F$-free…

Let $\mathcal P(n)$ denote the power set of $[n]$, ordered by inclusion, and let $\mathcal P (n,p)$ denote the random poset obtained from $\mathcal P(n)$ by retaining each element from $\mathcal P (n)$ independently at random with…