Related papers: On Saturated $k$-Sperner Systems

Let $G$ be a connected graph of order $n$, whose minimum vertex degree is at least $k$. A subset $S$ of vertices in $G$ is a $k$-tuple total dominating set if every vertex of $G$ is adjacent to at least $k$ vertices in $S$. The minimum…

Let $H$ be a fixed graph. A graph $G$ is called {\it $H$-saturated} if $H$ is not a subgraph of $G$ but the addition of any missing edge to $G$ results in an $H$-subgraph. The {\it saturation number} of $H$, denoted $sat(n,H)$, is the…

We define sparse saturated fusion systems and show that, for odd primes, sparse systems are constrained. This simplifies the proof of the Glauberman-Thompson p-nilpotency theorem for fusion systems and a related theorem of Stellmacher. We…

A subfamily $\mathcal{G}\subseteq \mathcal{F}\subseteq 2^{[n]}$ of sets is a non-induced (weak) copy of a poset $P$ in $\mathcal{F}$ if there exists a bijection $i:P\rightarrow \mathcal{G}$ such that $p\le_P q$ implies $i(p)\subseteq i(q)$.…

For a finite set $A\subset\mathbb{N}$ and $k\in \mathbb{N}$, let $\omega_k(A) = \sum_{i\in A, i\neq k}1$. For each $n\in \mathbb{N}$, define $$a_{k, n}\ =\ |\{E\subset \mathbb{N}\,:\, E = \emptyset\mbox{ or } \omega_k(E) < \min E\leqslant…

Recall that in a laminar family, any two sets are either disjoint or contained one in the other. Here, a parametrized weakening of this condition is introduced. Let us say that a set system $\mathcal{F} \subseteq 2^X$ is $t$-laminar if $A,B…

We investigate the combinatorial discrepancy of geometric set systems having bounded shallow cell complexity in the \emph{Beck-Fiala} setting, where each point belongs to at most $t$ ranges. For set systems with shallow cell complexity…

A $(k,m)$-Furstenberg set is a subset $S \subset \mathbb{F}_q^n$ with the property that each $k$-dimensional subspace of $\mathbb{F}_q^n$ can be translated so that it intersects $S$ in at least $m$ points. Ellenberg and Erman proved that…

Given a graph $G=(V,E)$ with non-negative real edge lengths and an integer parameter $k$, the Min-Max k-Tree Cover problem seeks to find a set of at most $k$ subtrees of $G$, such that the union of the trees is the vertex set $V$. The…

Suppose that $k\geq 2$ and $A$ is a non-empty subset of a finite abelian group $G$ with $|G|>1$. Then the cardinality of the restricted sumset $$ k^\wedge A:=\{a_1+\cdots+a_k:\,a_1,\ldots,a_k\in A,\ a_i\neq a_j\text{ for }i\neq j\} $$ is at…

The following classical question in extremal set theory is due to Erd\H os and S\'os: what is the size of the largest family $\mathcal F\subset {[n]\choose k}$ with no two sets $F_1,F_2\in \mathcal F$ such that $|F_1\cap F_2| = t$? In this…

For a given fixed poset $\mathcal P$ we say that a family of subsets of $[n]$ is $\mathcal P$-saturated if it does not contain an induced copy of $\mathcal P$, but whenever we add to it a new set, an induced copy of $\mathcal P$ is formed.…

We study subsets of $\mathbb{F}_p^n$ that do not contain progressions of length $k$. We denote by $r_k(\mathbb{F}_p^n)$ the cardinality of such subsets containing a maximal number of elements. In this paper we focus on the case $k=p$ and…

A $(k,m)$-Furstenberg set $S \subset \mathbb{F}_q^n$ over a finite field is a set that has at least $m$ points in common with a $k$-flat in every direction. The question of determining the smallest size of such sets is a natural…

Let $F$ be a strictly $1$-balanced $k$-graph on $s$ vertices with $t$ edges and $\delta_{F,d}^T$ be the infimum of $\delta>0$ such that for every $\alpha>0$ and sufficiently large $n\in \mathbb{N}$, every $k$-graph system $\mathbf…

For a graph $G=(V,E)$ and a set $S\subseteq V(G)$ of size at least $2$, an $S$-Steiner tree $T$ is a subgraph of $G$ that is a tree with $S\subseteq V(T)$. Two $S$-Steiner trees $T$ and $T'$ are internally disjoint (resp. edge-disjoint) if…

Let $\Pi_q$ be an arbitrary finite projective plane of order $q$. A subset $S$ of its points is called saturating if any point outside $S$ is collinear with a pair of points from $S$. Applying probabilistic tools we improve the upper bound…

Motivated by the work of Anstee, Griggs, and Sali on forbidden submatrices and the extremal sat-function for graphs, we introduce sat-type problems for matrices. Let F be a family of k-row matrices. A matrix M is called F-admissible if M…

A $k$-crossing family in a point set $S$ in general position is a set of $k$ segments spanned by points of $S$ such that all $k$ segments mutually cross. In this short note we present two statements on crossing families which are based on…

Let $F$ be an $n$-point set in $\mathbb{K}^d$ with $\mathbb{K}\in\{\mathbb{R},\mathbb{Z}\}$ and $d\geq 2$. A (discrete) X-ray of $F$ in direction $s$ gives the number of points of $F$ on each line parallel to $s$. We define…