Related papers: Uniform sets in a family with restricted intersect…

In this paper we study two directions of extending the classical Erd\H os-Ko-Rado theorem which states that any family of $k$-element subsets of the set $[n] = \{1,\ldots,n\}$ in which any two sets intersect, has cardinality at most…

Let $V$ be an $(n+\ell)$-dimensional vector space over a finite field, and $W$ a fixed $\ell$-dimensional subspace of $V$. Write ${V\brack n,0}$ to be the set of all $n$-dimensional subspaces $U$ of $V$ satisfying $\dim(U\cap W)=0$. A…

This paper resolves two open problems in extremal set theory. For a family $\mathcal{F} \subseteq 2^{[n]}$ and $i, j\in [n]$, we denote $\mathcal{F} (i,\bar{j})=\{F\backslash\{i\}: F\in \mathcal{F}, F\cap\{i,j\}=\{i\}\}$. The sturdiness…

Let $k$ and $m$ be positive integers. A collection of $k$-multisets from $\{1,..., m \}$ is intersecting if every pair of multisets from the collection is intersecting. We prove that for $m \geq k+1$, the size of the largest such collection…

For a given number of $k$-sets, how should we choose them so as to minimize the union-closed family that they generate? Our main aim in this paper is to show that, if $\mathcal{A}$ is a family of $k$-sets of size $\binom{t}{k}$, and $t$ is…

A $k$-partition of an $n$-set $X$ is a collection of $k$ pairwise disjoint non-empty subsets whose union is $X$. A family of $k$-partitions of $X$ is called $t$-intersecting if any two of its members share at least $t$ blocks. A…

Let $V$ be an $n$-dimensional vector space over the finite field $\mathbb{F}_q$ and ${V\brack k}$ denote the family of all $k$-dimensional subspaces of $V$. A family $\mathcal{F}\subseteq {V\brack k}$ is called $k$-uniform $r$-wise…

Let $V$ be a finite dimensional vector space over a finite field. Suppose that $\mathscr{F}_1$, $\mathscr{F}_2$, $\dots$, $\mathscr{F}_r$ are $r$-cross $t$-intersecting families of $k$-subspaces of $V$. In this paper, we determine the…

A central theme in extremal combinatorics is the study of the maximum number of edges in an $r$-uniform hypergraph ($r$-graph) with matching number at most $s$ (the Erd\H{o}s Matching Conjecture) or with pairwise intersection at least $t$…

The maximum size of $t$-intersecting families is one of the most celebrated topics in combinatorics, and its size is known as the Erd\H{o}s-Ko-Rado theorem. Such intersecting families, also known as constant-weight anticodes in coding…

Motivated by odd-sunflowers, introduced recently by Frankl, Pach, and P{\'a}lv{\"o}lgyi, we initiate the study of temperate families: a family $\mathcal{F} \subseteq \mathcal{P}([n])$ is said to be \emph{temperate} if each $A \in…

Families $\mathcal{A}_1, \mathcal{A}_2, ..., \mathcal{A}_k$ of sets are said to be \emph{cross-intersecting} if for any $i$ and $j$ in $\{1, 2, ..., k\}$ with $i \neq j$, any set in $\mathcal{A}_i$ intersects any set in $\mathcal{A}_j$. For…

For positive integers $n>k>t$ let $\binom{[n]}{k}$ denote the collection of all $k$-subsets of the standard $n$-element set $[n]=\{1,\ldots,n\}$. Subsets of $\binom{[n]}{k}$ are called $k$-graphs. A $k$-graph $\mathcal{F}$ is called…

An $(n, k_1, \dots, k_t)$-cross intersecting system is a set of non-empty pairwise cross-intersecting families $\mathcal{F}_1\subset{[n]\choose k_1}, \mathcal{F}_2\subset{[n]\choose k_2}, \dots, \mathcal{F}_t\subset{[n]\choose k_t}$ with…

We consider a problem of maximizing the product of the sizes of two uniform cross-$t$-intersecting families of sets. We show that the value of this maximum is at most polynomially larger (in the size of a ground set) than a quantity…

To study how balanced or unbalanced a maximal intersecting family $\mathcal{F}\subseteq \binom{[n]}{r}$ is we consider the ratio $\mathcal{R}(\mathcal{F})=\frac{\Delta(\mathcal{F})}{\delta(\mathcal{F})}$ of its maximum and minimum degree.…

Let M be a subset of {0, .., n} and F be a family of subsets of an n element set such that the size of A intersection B is in M for every A, B in F. Suppose that l is the maximum number of consecutive integers contained in M and n is…

A family X of sets is said to be intersecting if any two members of X have non-empty intersection. It is a well-known and simple fact that an intersecting family of subsets of [n]={1,2,...,n} can contain at most 2^(n-1) sets. Katona, Katona…

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 $X$…

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…