Related papers: Bisecting and D-secting families for set systems

We say that a set system $\mathcal{F}\subseteq 2^{[n]}$ shatters a given set $S\subseteq [n]$ if $2^S= \{F~\cap~S:~F~\in~\mathcal{F}\}$. The Sauer-Shelah lemma states that in general, a set system $\mathcal{F}$ shatters at least…

A family $\mathcal{F}$ of $k$-subsets of an $n$-set is called $s$-almost $t$-intersecting if each member is $t$-disjoint with at most $s$ members. In this paper, we prove that, if $\left|\mathcal{F}\right|$ is maximum, then $\mathcal{F}$…

We determine the finite groups $G$ in which every subset $A \subseteq G$ of cardinality dividing the order of $G$ is a \emph{factor}, i.e. has a complement $B \subseteq G$ of cardinality $|G|/|A|$ such that $G = A \cdot B$ or $G = B \cdot…

We consider $k$-graphs on $n$ vertices, that is, $\mathcal{F}\subset \binom{[n]}{k}$. A $k$-graph $\mathcal{F}$ is called intersecting if $F\cap F'\neq \emptyset$ for all $F,F'\in \mathcal{F}$. In the present paper we prove that for $k\geq…

Given a set $\mathcal{F}$ of graphs, a graph $G$ is $\mathcal{F}$-free if $G$ does not contain any member of $\mathcal{F}$ as an induced subgraph. Barrus, Kumbhat, and Hartke [M. D. Barrus, M. Kumbhat, and S. G. Hartke, Graph classes…

We study families of subsets of $\omega$ which are independent with respect to the asymptotic density $\mathsf{d}$. We show, for instance, that there exists a maximal $\mathsf{d}$-independent family $\mathcal{A}$ such that…

For every integer $k \geq 2$ and every $A \subseteq \mathbb{N}$, we define the \emph{$k$-directions sets} of $A$ as $D^k(A) := \{{\bf a} / \|{\bf a}\| : {\bf a} \in A^k\}$ and $D^{\underline{k}}(A) := \{{\bf a} / \|{\bf a}\| : {\bf a} \in…

Let $X$ be an $n$-element set, where $n$ is even. We refute a conjecture of J. Gordon and Y. Teplitskaya, according to which, for every maximal intersecting family $\mathcal{F}$ of $\frac{n}2$-element subsets of $X$, one can partition $X$…

For a property $\Gamma$ and a family of sets $\cF$, let $f(\cF,\Gamma)$ be the size of the largest subfamily of $\cF$ having property $\Gamma$. For a positive integer $m$, let $f(m,\Gamma)$ be the minimum of $f(\cF,\Gamma)$ over all…

A subset of vertices $F$ in a graph $G$ is called a \emph{dissociation set} if the induced subgraph $G[F]$ of $G$ has maximum degree at most 1. A \emph{maximal dissociation set} of $G$ is a dissociation set which is not a proper subset of…

A family of sets $\mathcal{F} \subseteq 2^{[n]}$ is defined to be $l$-trace $k$-Sperner if for any $l$-subset $L$ of $[n]$ the family of traces $\mathcal{F}|_L=\{F \cap L: F \in \mathcal{F}\}$ does not contain any chain of length $k+1$. In…

Let $X$ be a set, $B_{X}$ denotes the family of all subsets of $X$ and $F: X \longrightarrow B_{X}$ be a set-valued mapping such that $x \in F(x)$, $sup_{x\in X} | F(x)|< \kappa$, $sup_{x\in X} | F^{-1}(x)|< \kappa$ for all $x\in X$ and…

We define a $C(k)$ to be a family of $k$ sets $F_1,\dots,F_k$ such that $\textrm{conv}(F_i\cup F_{i+1})\cap \textrm{conv}(F_j\cup F_{j+1})=\emptyset$ when $\{i,i+1\}\cap \{j,j+1\}=\emptyset$ (indices are taken modulo $k$). We show that if…

A family of sets $\mathcal{A}$ is union-closed if it is finite and nonempty with member sets that are all finite and distinct (at least one of which is nonempty) and it satisfies the property $X, Y \in \mathcal{A} \implies X \cup Y \in…

We consider families F of sequences converging to +infinity that F satisfies the following condition (C): (C): if an open set U in the real line is unbounded above then there exists a sequence belonging to F, which has an infinite number of…

A family of subsets of a $t$-set is a \emph{$d$-cover-free family} or $d$-CFF if no subset in the family is contained in the union of any $d$ other subsets. Let $t(d, n)$ denote the minimum $t$ for which there exists a $d$-CFF on a $t$-set…

Given an irreducible fraction $\frac{c}{d} \in [0,1]$, a pair $(\mathcal{A},\mathcal{B})$ is called a $\frac{c}{d}$-cross-intersecting pair of $2^{[n]}$ if $\mathcal{A}, \mathcal{B}$ are two families of subsets of $[n]$ such that for every…

Following D. Sobota we call a family $\mathcal F$ of infinite subsets of $\mathbb N$ a Rosenthal family if it can replace the family of all infinite subsets of $\mathbb N$ in classical Rosenthal's Lemma concerning sequences of measures on…

A set family ${\cal F}$ is called intersecting if every two members of ${\cal F}$ intersect, and it is called uniform if all members of ${\cal F}$ share a common size. A uniform family ${\cal F} \subseteq \binom{[n]}{k}$ of $k$-subsets of…

We say that a set system $\mathcal{F}$ is $k$-completely hyperseparating if for any vertex $v$, there are at most $k$ sets in $\mathcal{F}$ with intersection $\{v\}$. We determine the minimum size of such set systems on an $n$-element…