Related papers: Non-trivial $t$-intersecting families for vector s…

Consider a family $\mathcal{F}$ of $k$-subsets of an ambient $(k^2-k+1)$-set such that no pair of $k$-subsets in $\mathcal{F}$ intersects in exactly one element. In this short note we show that the maximal size of such $\mathcal{F}$ is…

A family $\mathcal{F}$ of subsets of $\{1,2,\ldots,n\}$ is called a $t$-intersecting family if $|F\cap G| \geq t$ for any two members $F, G \in \mathcal{F}$ and for some positive integer $t$. If $t=1$, then we call the family $\mathcal{F}$…

Ever since the famous Erd\H{o}s-Ko-Rado theorem initiated the study of intersecting families of subsets, extremal problems regarding intersecting properties of families of various combinatorial objects have been extensively investigated.…

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…

The classical Erd\H{o}s-Ko-Rado theorem on the size of an intersecting family of $k$-subsets of the set $[n] = \{1, 2, \dots, n\}$ is one of the fundamental intersection theorems for set systems. After the establishment of the EKR theorem,…

In this paper we study a question related to the classical Erd\H{o}s-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 $ n\geqslant t\geqslant 1$ and $ \mathcal{A}_1, \mathcal{A}_2, \ldots, \mathcal{A}_m \subseteq 2^{[n]}$ be non-empty families. We say that they are pairwise cross $t$-intersecting if $|A_i\cap A_j|\geqslant t$ holds for any $A_i\in…

A family of subsets of $[n]$ is $r$-wise agreeing if for any $r$ sets from the family there is an element $x$ that is either contained in all or contained in none of the $r$ sets. The study of such families is motivated by questions in…

In this paper, by shifting technique we study $t$-intersecting families for direct products where the ground set is divided into several parts. Assuming the size of each part is sufficiently large, we determine all extremal $t$-intersecting…

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 call a family of sets intersecting, if any two sets in the family intersect. In this paper we investigate intersecting families $\mathcal{F}$ of $k$-element subsets of $[n]:=\{1,\ldots, n\},$ such that every element of $[n]$ lies in the…

We prove that if a subset of the d-dimensional vector space over a finite field is large enough, then it contains many k-tuples of mutually orthogonal vectors.

Let $[n]$ (resp. $V$) be an $n$-element set (resp. $n$-dimensional vector space over the finite field $\mathbb{F}_{q}$), and $\binom{[n]}{k}$ (resp. $\genfrac{[}{]}{0pt}{}{V}{k}$) denote the set of all $k$-subsets of $[n]$ (resp.…

For $n > 2k \geq 4$ we consider intersecting families $\mathcal F$ consisting of $k$-subsets of $\{1, 2, \ldots, n\}$. Let $\mathcal I(\mathcal F)$ denote the family of all distinct intersections $F \cap F'$, $F \neq F'$ and $F, F'\in…

Given a sequence of positive integers $p = (p_1, . . ., p_n)$, let $S_p$ denote the family of all sequences of positive integers $x = (x_1,...,x_n)$ such that $x_i \le p_i$ for all $i$. Two families of sequences (or vectors), $A,B \subseteq…

We introduce a measure for subspaces of a vector space over a $q$-element field, and propose some extremal problems for intersecting families. These are $q$-analogue of Erd\H{o}s-Ko-Rado type problems, and we answer some of the basic…

Given a finite abelian group $G$ and a subset $J\subset G$ with $0\in J$, let $D_{G}(J,N)$ be the maximum size of $A\subset G^{N}$ such that the difference set $A-A$ and $J^{N}$ have no non-trivial intersection. Recently, this extremal…

The celebrated Erd\H{o}s-Ko-Rado theorem \cite{EKR1961} states that the maximum intersecting $k$-uniform family on $[n]$ is a full star if $n\ge 2k+1$. Furthermore, Hilton-Milner \cite{HM1967} showed that if an intersecting $k$-uniform…

A subset $Y$ of the general linear group $\operatorname{GL}(n,q)$ is called $t$-intersecting if $\operatorname{rk}(x-y)\le n-t$ for all $x,y\in Y$, or equivalently $x$ and $y$ agree pointwise on a $t$-dimensional subspace of…

A covering number of a family is the size of the smallest set that intersects all sets from the family. In 1978 Frankl determined for $n\ge n_0(k)$ the largest intersecting family of $k$-element subsets of $[n]$ with covering number $3$. In…