Related papers: Repeated columns and an old chestnut

In 2001, Woodall conjectured that for every pair of integers $s,t \ge 1$, all graphs without a $K_{s,t}$-minor are $(s+t-1)$-choosable. In this note we refute this conjecture in a strong form: We prove that for every choice of constants…

We prove that for all positive integers $n$ and $k$, there exists an integer $N = N(n,k)$ satisfying the following. If $U$ is a set of $k$ direction vectors in the plane and $\mathcal{J}_U$ is the set of all line segments in direction $u$…

A family $\mbox{$\cal F$}=\{F_1,\ldots,F_m\}$ of subsets of $[n]$ is said to be ordered, if there exists an $1\leq r\leq m$ index such that $n\in F_i$ for each $1\leq i\leq r$, $n\notin F_i$ for each $i>r$ and $|F_i|\leq |F_j|$ for each…

Let $m$, $n$, and $k$ be integers satisfying $0 < k \leq n < 2k \leq m$. A family of sets $\mathcal{F}$ is called an $(m,n,k)$-intersecting family if $\binom{[n]}{k} \subseteq \mathcal{F} \subseteq \binom{[m]}{k}$ and any pair of members of…

Let $V=\oC^n$ and let $T:=T(V)\otimes T(V^*)$ be the mixed tensor algebra over $V$. We characterize those subsets $A$ of $T$ for which there is a subgroup $G$ of the unitary group $\UU(n)$ such that $A=T^G$. They are precisely the…

Let $t$ be a non-negative integer and $\mbox{$\cal P$}=\{(A_i,B_i)\}_{1\leq i\leq m}$ be a set-pair family satisfying $|A_i \cap B_i|\leq t$ for $1\leq i \leq m$. $\mbox{$\cal P$}$ is called strong Bollob\'as $t$-system, if $|A_i\cap…

Let $ k, m, n $ be positive integers with $ k \geq 2 $. A $ k $-multiset of $ [n]_m $ is a collection of $ k $ integers from the set $ \{1, 2, \ldots, n\} $ in which the integers can appear more than once but at most $ m $ times. A family…

Let $\F$ be a finite family of axis-parallel boxes in $\R^d$ such that $\F$ contains no $k+1$ pairwise disjoint boxes. We prove that if $\F$ contains a subfamily $\M$ of $k$ pairwise disjoint boxes with the property that for every $F\in \F$…

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…

Given two numerical semigroups $S$ and $T$ we say that $T$ is a multiple of $S$ if there exists an integer $d \in \mathbb{N} \setminus \{0\}$ such that $S = \{x \in \mathbb{N} \mid d x \in T\}$. In this paper we study the family of…

The extension of pattern avoidance from ordinary permutations to those on multisets gave birth to several interesting enumerative results. We study permutations on regular multisets, i.e., multisets in which each element occurs the same…

We prove that the Lie algebra $\mathfrak{sl}_n(\textbf{F}_q)$ of traceless matrices over a finite field of characteristic $p$ can be generated by $2$ elements with exceptions when $(n, p)$ is $(3, 3)$ or $(4,2)$. In the latter cases, we…

A set-family ${\cal F}$ is disjointness-compliable if $A' \subseteq A \in {\cal F}$ implies $A' \in {\cal F}$ or $A \setminus A' \in {\cal F}$; if ${\cal F}$ is also symmetric then ${\cal F}$ is proper. A classic result of Goemans and…

Let $S\subseteq \mathbb{N}$ be a numerical semigroup with multiplicity $m$, embedding dimension $\nu$ and conductor $c=f+1=qm-\rho$ for some $q,\rho\in\mathbb{N}$ with $\rho<m$. Let Ap$(S,m) = \{w\_0<w_1 < \ldots < w_{m-1}\}$ be the Ap\'ery…

Let $ k, n \in \mathbb{N}^+ $ and $ m \in \mathbb{N}^+ \cup \{\infty \} $. A $ k $-multiset in $ [n]_m $ is a $ k $-set whose elements are integers from $ \{1, 2, \ldots, n\} $, and each element is allowed to have at most $ m $ repetitions.…

The author, together with Nagy, studied the following problem on unavoidable intersections of given size in binary affine spaces. Given an $m$-element set $S\subseteq \mathbb{F}_2^n$, is there guaranteed to be a $[k,t]$-flat, that is, a…

A set system $\mathcal{F}$ is $t$-\textit{intersecting}, if the size of the intersection of every pair of its elements has size at least $t$. A set system $\mathcal{F}$ is $k$-\textit{Sperner}, if it does not contain a chain of length…

In this paper, we determine the largest family $\mathcal F \subset 2^{[n]}$ without $s$ pairwise disjoint sets, provided $n=ms+c$ for positive integers $m,c$, and $s \geq s_0(m, c)$. This result can be seen as a non-uniform analogue of the…

The number of $n \times n$ matrices whose entries are either -1, 0, or 1, whose row- and column- sums are all 1, and such that in every row and every column the non-zero entries alternate in sign, is proved to be $[1!4! >...…

We show that for any finitely generated group of matrices that is not virtually solvable, there is an integer m such that, given an arbitrary finite generating set for the group, one may find two elements a and b that are both products of…