Related papers: Fractional L-intersecting 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 $V$. The families $\mathcal{F}_1\subseteq{V\brack k_1},\mathcal{F}_2\subseteq{V\brack…

A family of sets is said to be \emph{intersecting} if any two sets in the family have nonempty intersection. In 1973, Erd\H{o}s raised the problem of determining the maximum possible size of a union of $r$ different intersecting families of…

Let $[n]:=\lbrace 1,2,\ldots,n \rbrace$, and $M$ be a set of positive integers. Denote the family of all subsets of $[n]$ with sizes in $M$ by $\binom{\left[n\right]}{M}$. The non-empty families…

For an integer $d \geq 2$, a family $\mathcal{F}$ of sets is $\textit{$d$-wise intersecting}$ if for any distinct sets $A_1,A_2,\dots,A_d \in \mathcal{F}$, $A_1 \cap A_2 \cap \dots \cap A_d \neq \emptyset$, and $\textit{non-trivial}$ if…

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…

Using the sunflower method, we show that if $\theta \in (0,1) \cap \mathbb{Q}$ and $\mathcal{F}$ is a $O(n^{1/3})$-bounded $\theta$-intersecting family over $[n]$, then $\lvert \mathcal{F} \rvert = O(n)$, and that if $\mathcal{F}$ is…

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…

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…

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 $F$ 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…

Given a family $\mathcal{F}\subset 2^{[n]}$ and $1\leq i\neq j\leq n$, we use $\mathcal{F}(\bar{i},j)$ to denote the family $\{F\setminus \{j\}\colon F\in \mathcal{F},\ F\cap \{i,j\}=\{j\}\}$. The sturdiness of $\mathcal{F}$ is defined as…

The families $\mathcal F_1\subseteq \binom{[n]}{k_1},\mathcal F_2\subseteq \binom{[n]}{k_2},\dots,\mathcal F_r\subseteq \binom{[n]}{k_r}$ are said to be cross-intersecting if $|F_i\cap F_j|\geq 1$ for any $1\leq i<j\leq r$ and $F_i\in…

Two families $\mathcal{F}$ and $\mathcal{G}$ are called cross-intersecting if for every $F\in \mathcal{F}$ and $G\in \mathcal{G}$, the intersection $F\cap G$ is non-empty. It is significant to determine the maximum sum of sizes of…

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…

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 $\vert\bigcap_{i\in[r]}F_i\vert\geq t$.…

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 different extremal properties of intersecting families, as well as the structure of large intersecting…

A family $\mathcal{F}\subset \binom{[n]}{k}$ is called an intersecting family if $F\cap F'\neq \emptyset$ for all $F,F'\in \mathcal{F}$. If $\cap \mathcal{F}\neq \emptyset$ then $\mathcal{F}$ is called a star. The diversity of an…

Given a finite set $X$ of points in $R^n$ and a family $F$ of sets generated by the pairs of points of $X$, we determine volumetric and structural conditions for the sets that allow us to guarantee the existence of a positive-fraction…

Two families $\mathcal{A}$ and $\mathcal{B}$ are cross-intersecting if $A\cap B\ne \emptyset$ for any $A\in \mathcal{A}$ and $B\in \mathcal{B}$. We call $t$ families $\mathcal{A}_1, \mathcal{A}_2,\dots, \mathcal{A}_t$ pairwise…

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…