Related papers: Non-trivial 3-wise intersecting uniform families

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…

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 $\mathcal{F}$…

Let $n$, $r$, $k_1,\ldots,k_r$ and $t$ be positive integers with $r\geq 2$, and $\mathcal{F}_i\ (1\leq i\leq r)$ a family of $k_i$-subsets of an $n$-set $V$. The families $\mathcal{F}_1,\ \mathcal{F}_2,\ldots,\mathcal{F}_r$ are said to be…

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…

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 study the following natural arithmetic question regarding intersecting families: how large can a family of subsets of integers from $\{1, \ldots n\}$ be such that, for every pair of subsets in the family, the intersection contains a sum…

Simonovits and S\'{o}s conjectured that the maximal size of a triangle-intersecting family of graphs on $n$ vertices is $2^{\binom{n}{2}-3}$. Their conjecture has recently been proved using spectral methods. We provide an elementary proof…

Guo and Xu determined the maximum size of intersecting families over finite affine spaces and showed that any family reaches maximum size must be trivial. In this paper, we characterize non-trivial intersecting family with maximum size.

The families $\mathcal{A}$ and $\mathcal{B}$ are cross intersecting if $A\cap B\ne \emptyset$ for any $A\in \mathcal{A}$ and $B\in \mathcal{B}$. Let $t\geq 2$ and $k_1\geq k_2\geq \cdots \geq k_t$. We say that $(\mathcal{F}_1, \dots,…

Two families $\mathcal{A}$ and $\mathcal{B}$ of sets are said to be cross-intersecting if each member of $\mathcal{A}$ intersects each member of $\mathcal{B}$. For any two integers $n$ and $k$ with $0 \leq k \leq n$, let ${[n] \choose \leq…

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 $\mathcal{F}$…

The celebrated Erd\H{o}s-Ko-Rado theorem determines the maximum size of a $k$-uniform intersecting family. The Hilton-Milner theorem determines the maximum size of a $k$-uniform intersecting family that is not a subfamily of the so-called…

We say that a set $A$ $t$-intersects a set $B$ if $A$ and $B$ have at least $t$ common elements. Two families $\mathcal{A}$ and $\mathcal{B}$ of sets are said to be cross-$t$-intersecting if each set in $\mathcal{A}$ $t$-intersects each set…

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…

For a family $\mathcal{F}$ of subsets of a finite set, define $\mathcal{D}(\mathcal{F})=\{F\setminus F': F, F'\in\mathcal{F}\}$. A family $\mathcal{F}$ is called intersecting if $F\cap F'\not=\emptyset$ for all $F, F'\in\mathcal{F}$. Frankl…

A family of graphs F is said to be triangle-intersecting if for any two graphs G,H in F, the intersection of G and H contains a triangle. A conjecture of Simonovits and Sos from 1976 states that the largest triangle-intersecting families of…

In this note, we give short proofs of three theorems about intersection problems. The first one is a determination of the maximum size of a nontrivial $k$-uniform, $d$-wise intersecting family for $n\ge \left(1+\frac{d}{2}\right)(k-d+2)$,…

We say that a family of $k$-subsets of an $n$-element set is intersecting if any two of its sets intersect. In this paper we study properties and structure of large intersecting families. We prove a conclusive version of Frankl's theorem on…

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…

A family $\mathcal F\subset 2^{[n]}$ is called intersecting if any two of its sets intersect. Given an intersecting family, its diversity is the number of sets not passing through the most popular element of the ground set. Peter Frankl…