Related papers: An $(\aleph_0,k+2)$-Theorem for $k$-Transversals

We provide answers to a question brought up by Erd\H{o}s about the construction of Wetzel families in the absence of the continuum hypothesis - a Wetzel family is a family $\mathcal{F}$ of entire functions on the complex plane which…

Let $n$, $r$, $k_1,\ldots,k_r$ and $t$ be positive integers with $r\geq 2$, and $\mathcal{F}_i\ (1\leq i\leq r)$ a family of $k_i$-subsets of an $n$-set $V$. The families $\mathcal{F}_1,\ \mathcal{F}_2,\ldots,\mathcal{F}_r$ are said to be…

Let $\mathcal{S}$ be a family of sets with VC-codensity less than $2$. We prove that, if $\mathcal{S}$ has the $(\omega, 2)$-property (for any infinitely many sets in $\mathcal{S}$, at least $2$ among them intersect), then $\mathcal{S}$ can…

In 1997 we proved that any triangular semisimple Hopf algebra over an algebraically closed field k of characteristic 0 is obtained from the group algebra k[G] of a finite group G, by twisting its comultiplication by a twist in the sense of…

We show that under certain conditions, well-studied algebraic properties transfer from the class $\mathcal{Q}_{_\text{RFSI}}$ of the relatively finitely subdirectly irreducible members of a quasivariety $\mathcal{Q}$ to the whole…

Ahlswede and Khachatrian's diametric theorem is a weighted version of their complete intersection theorem, itself an extension of the $t$-intersecting Erd\H{o}s-Ko-Rado theorem. Their intersection theorem says that the maximum size of a…

We compare two methods of proving separable reduction theorems in functional analysis -- the method of rich families and the method of elementary submodels. We show that any result proved using rich families holds also when formulated with…

It is proved that if there exists a positive and continuous function $f$ on an $n$-dimensional complex manifold $X$, $q$-convex with corners outside a compact set $K\subset X$ and which exhausts $X$ from below, then…

What use can there be for a function from the $p$-adic numbers to the $q$-adic numbers, where $p$ and $q$ are distinct primes? The traditional answer, courtesy of the half-century old theory of non-archimedean functional analysis: not much.…

A graph of order $n$ is said to be $k$-\emph{factor-critical} $(0\le k<n)$ if the removal of any $k$ vertices results in a graph with a perfect matching. A $k$-factor-critical graph $G$ is \emph{minimal} if $G-e$ is not $k$-factor-critical…

Let X and Y be finite dimensional normed spaces, F(X,Y) a collection of all mappings from X into Y. A mapping $P\in F(X,Y)$ is said to be piecewise affine if there exists a finite family of convex polyhedral subsets covering X and such that…

The long-standing topological Tverberg conjecture claimed, for any continuous map from the boundary of an $N(q,d):=(q-1)(d+1)$-simplex to $d$-dimensional Euclidian space, the existence of $q$ pairwise disjoint subfaces whose images have…

For a permutation f of an n-dimensional vector space V over a finite field of order q we let k-affinity(f) denote the number of k-flats X of V such that f(X) is also a k-flat. By k-spectrum(n,q) we mean the set of integers k-affinity(f)…

A disjoint $(v,k,k-1)$ difference family in an additive group $G$ is a partition of $G\setminus\{0\}$ into sets of size $k$ whose lists of differences cover, altogether, every non-zero element of $G$ exactly $k-1$ times. The main purpose of…

Two families $\mathcal{A}$ and $\mathcal{B}$ of sets are said to be cross-intersecting if each member of $\mathcal{A}$ intersects each member of $\mathcal{B}$. For any two integers $n$ and $k$ with $0 \leq k \leq n$, let ${[n] \choose \leq…

In this paper, we will answer one of the questions proposed by Bodur, Del~Pia, Dey, Molinaro and Pokutta in 2017. Specifically, we show that the k-aggregation closure of a covering set is a polyhedron. The proof technique is based on an…

For $n\leq d$, a family ${\cal F}=\{C_0,C_1,\ldots, C_n\}$ of compact convex sets in $R^d$ is called an $n$-critical family provided any $n$ members of ${\cal F}$ have a non-empty intersection, but $\bigcap_{i=0}^n C_i=\varnothing$. If…

Let $d\geq 2$ be a positive integer, $K$ an algebraically closed field of characteristic not dividing $d$, $n\geq d+1$ a positive integer that is prime to $d$, $f(x)\in K[x]$ a degree $n$ monic polynomial without multiple roots, $C_{f,d}:…

This article contains a proof of the MDS conjecture for $k \leq 2p-2$. That is, that if $S$ is a set of vectors of ${\mathbb F}_q^k$ in which every subset of $S$ of size $k$ is a basis, where $q=p^h$, $p$ is prime and $q$ is not and $k \leq…

In this paper we study some generalized versions of a recent result due to Covert, Koh, and Pi (2015). More precisely, we prove that if a subset $\mathcal{E}$ in a regular variety satisfies $|\mathcal{E}|\gg…