Related papers: Some intersection theorems for finite sets

Set systems with strongly restricted intersections, called $\alpha$-intersecting families for a vector $\alpha$, were introduced recently as a generalization of several well-studied intersecting families including the classical oddtown and…

A family $\mathcal F\subset {[n]\choose k}$ is $U(s,q)$ of for any $F_1,\ldots, F_s\in \mathcal F$ we have $|F_1\cup\ldots\cup F_s|\le q$. This notion generalizes the property of a family to be $t$-intersecting and to have matching number…

We say that a family of $k$-subsets of an $n$-element set is {\it intersecting}, if any two of its sets intersect. In this paper, we study the structure of large intersecting families. Several years ago, Han and Kohayakawa (Proc. AMS,…

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…

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…

A family $F$ of graphs on a fixed set of $n$ vertices is called triangle-intersecting if for any $G_1,G_2 \in F$, the intersection $G_1 \cap G_2$ contains a triangle. More generally, for a fixed graph $H$, a family $F$ is $H$-intersecting…

Let $\mathcal A=\{A_1,\ldots,A_n\}$ be a family of sets in the plane. For $0 \leq i < n$, denote by $f_i$ the number of subsets $\sigma$ of $\{1,\ldots,n\}$ of cardinality $i+1$ that satisfy $\bigcap_{i \in \sigma} A_i \neq \emptyset$. Let…

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.

Let $\mathcal{A}\subseteq{[n]\choose a}$ and $\mathcal{B}\subseteq{[n]\choose b}$ be two families of subsets of $[n]$, we say $\mathcal{A}$ and $\mathcal{B}$ are cross-intersecting if $A\cap B\neq \emptyset$ for all $A\in\mathcal{A}$,…

Two sets $\mathscr{A}$ and $\mathscr{B}$ are said to be cross-intersecting if $X\cap Y\neq\emptyset$ for all $X\in\mathscr{A}$ and $Y\in\mathscr{B}$. Given two cross-intersecting Sperner families (or antichains) $\mathscr{A}$ and…

Two families of sets \(\mathcal{A}\) and \(\mathcal{B}\) are called \emph{cross-\(t\)-intersecting} if \(|A \cap B| \geq t\) for all \(A \in \mathcal{A}\) and \(B \in \mathcal{B}\). Determining the maximum product of sizes for such…

The profile vector of a family $\mathcal{F}$ of subsets of an $n$-element set is $(f_0,f_1, \ldots, f_n)$ where $f_i$ denotes the number of the $i$-element members of $\mathcal{F}$. In this paper we determine the extreme points of the set…

Mubayi's Conjecture states that if $\mathcal{F}$ is a family of $k$-sized subsets of $[n] = \{1,\ldots,n\}$ which, for $k \geq d \geq 2$, satisfies $A_1 \cap\cdots\cap A_d \neq \emptyset$ whenever $|A_1 \cup\cdots\cup A_d| \leq 2k$ for all…

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…

A family $\F$ of sets is said to be intersecting if any two sets in $\F$ have nonempty intersection. The celebrated Erd{\H o}s-Ko-Rado theorem determines the size and structure of the largest intersecting family of $k$-sets on an $n$-set…

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 $n\ge \left(1+\frac{d}{2}\right)(k-d+2)$,…

Let $k, r, n \geq 1$ be integers, and let $\S_{n, k, r}$ be the family of $r$-signed $k$-sets on $[n] = \{1, \dots, n\}$ given by $$ \mathcal{S}_{n, k, r} = \Big\{\{(x_1, a_1), \dots, (x_k, a_k)\}: \{x_1, \dots, x_k\} \in \binom{[n]}{k},…

Let $\omega(\mathcal{F})=\sum_{\{A,B\}\subset\mathcal{F}}|A\cap B|$ and $\omega(\mathcal{A},\mathcal{B})=\sum_{(A,B)\in \mathcal{A}\times \mathcal{B}}|A\cap B|$. A family $\mathcal{F}$ is intersecting if $F_1\cap F_2\neq \emptyset$ for any…

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…

We say a family of sets is intersecting if any two of its sets intersect, and we say it is trivially intersecting if there is an element which appears in every set of the family. In this paper we study the maximum size of a non-trivially…