Related papers: On generating sets of left-compressed intersecting…

A set $\mathcal{F} \subset \operatorname{Sym}(V)$ is \textsl{intersecting} if any two of its elements agree on some element of $V$. Given a finite transitive permutation group $G\leq \operatorname{Sym}(V)$, the \textsl{intersection density}…

Let $2^{[n]}$ and $\binom{[n]}{i}$ be the power set and the class of all $i$-subsets of $\{1,2,\cdots,n\}$, respectively. We call two families $\mathscr{A}$ and $\mathscr{B}$ cross-intersecting if $A\cap B\neq \emptyset$ for any $A\in…

Let $\mathcal F\subset 2^{[n]}$ be a family in which any three sets have non-empty intersection and any two sets have at least $38$ elements in common. The nearly best possible bound $|\mathcal F|\le 2^{n-2}$ is proved. We believe that $38$…

We construct generating trees with one, two, and three labels for some classes of permutations avoiding generalized patterns of length 3 and 4. These trees are built by adding at each level an entry to the right end of the permutation,…

In this paper, we investigate two questions on Kneser graphs $KG_{n,k}$. First, we prove that the union of $s$ intersecting families in ${[n]\choose k}$ has size at most ${n\choose k}-{n-s\choose k}$ for all sufficiently large $n$ that…

For $k\geq3$, a collection of $k$ sets is said to form a \emph{weak $\Delta$-system} if the intersection of any two sets from the collection has the same size. Erd\H{o}s and Szemer\'{e}di asked about the size of the largest family…

A family $\mathcal{F}$ of spanning trees of the complete graph on $n$ vertices $K_n$ is \emph{$t$-intersecting} if any two members have a forest on $t$ edges in common. We prove an Erd\H{o}s--Ko--Rado result for $t$-intersecting families of…

A family of $k$-subsets $A_1, A_2, ..., A_d$ on $[n]=\{1,2,..., n\}$ is called a $(d, c)$-cluster if the union $A_1\cup A_2 \cup ... \cup A_d$ contains at most $ck$ elements with $c<d$. Let $\mathcal{F}$ be a family of $k$-subsets of an…

Let $\mathcal F\subset 2^{[n]}$ be an $s$-uniform family such that every two distinct sets have a nonempty intersection but intersect in at most $k$ elements. By the well-known Ray-Chaudhuri--Wilson theorem, since the intersections can take…

Let $G$ be a finite group. In order to determine the smallest cardinality $d(G)$ of a generating set of $G$ and a generating set with this cardinality, one should repeat many times the test whether a subset of $G$ of small cardinality…

Two families $\mathcal A\subseteq\binom{[n]}{k}$ and $\mathcal B\subseteq\binom{[n]}{\ell}$ are called cross-$t$-intersecting if $|A\cap B|\geq t$ for all $A\in\mathcal A$, $B\in\mathcal B$. Let $n$, $k$ and $\ell$ be positive integers such…

A finite family $\mathrsfs{F}$ of subsets of a finite set $X$ is union-closed whenever $f,g\in\mathrsfs{F}$ implies $f\cup g\in\mathrsfs{F}$. These families are well known because of Frankl's conjecture. In this paper we developed further…

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…

Many groups possess highly symmetric generating sets that are naturally endowed with an underlying combinatorial structure. Such generating sets can prove to be extremely useful both theoretically in providing new existence proofs for…

We give a simple argument showing that the number of edges in the intersection graph $G$ of a family of $n$ sets in the plane with a linear union-complexity is $O(\omega(G)n)$. In particular, we prove $\chi(G)\leq \text{col}(G)<…

We give a sufficient combinatorial condition for the non-negativity of the coefficients of polynomial quotients of products of $q$-integers, also known as cyclotomic generating functions (CGFs). This slightly extends work by Iano-Fletcher,…

Given integers $r\geq 2$ and $n,t\geq 1$ we call families $\mathcal{F}_1,\dots,\mathcal{F}_r\subseteq\mathscr{P}([n])$ $r$-cross $t$-intersecting if for all $F_i\in\mathcal{F}_i$, $i\in[r]$, we have $\vert\bigcap_{i\in[r]}F_i\vert\geq t$.…

A set of permutations $I \subset S_n$ is said to be {\em k-intersecting} if any two permutations in $I$ agree on at least $k$ points. We show that for any $k \in \mathbb{N}$, if $n$ is sufficiently large depending on $k$, then the largest…

Let $m\geq 2$, $n$ be positive integers, and $R_i=\{k_{i,1} >k_{i,2} >\cdots> k_{i,t_i}\}$ be subsets of $[n]$ for $i=1,2,\ldots,m$. The families $\mathcal{F}_1\subseteq \binom{[n]}{R_1},\mathcal{F}_2\subseteq…

In this paper, we derive a tight upper bound for the size of an intersecting $k$-Sperner family of subspaces of the $n$-dimensional vector space $\mathbb{F}_{q}^{n}$ over finite field $\mathbb{F}_{q}$ which gives a $q$-analogue of the…