Triangle-Intersecting Families of Graphs

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 graphs on a fixed set of n vertices are those obtained by fixing a specific triangle and taking all graphs containing it, resulting in a family of size (1/8) 2^{n choose 2}. We prove this conjecture and some generalizations (for example, we prove that the same is true of odd-cycle-intersecting families, and we obtain best possible bounds on the size of the family under different, not necessarily uniform, measures). We also obtain stability results, showing that almost-largest triangle-intersecting families have approximately the same structure.Comment: 43 page

A note on distinct differences in tt-intersecting families

For a family $\mathcal{F}$ of subsets of $\{1,2,\ldots,n\}$, let $\mathcal{D}(\mathcal{F}) = \{F\setminus G: F, G \in \mathcal{F}\}$ be the collection of all (setwise) differences of $\mathcal{F}$. The family $\mathcal{F}$ is called a $t$-intersecting family, if for some positive integer $t$ and any two members $F, G \in \mathcal{F}$ we have $|F\cap G| \geq t$. The family $\mathcal{F}$ is simply called intersecting if $t=1$. Recently, Frankl proved an upper bound on the size of $\mathcal{D}(\mathcal{F})$ for the intersecting families $\mathcal{F}$. In this note we extend the result of Frankl to $t$-intersecting families

An improved threshold for the number of distinct intersections of intersecting families

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 $\mathcal{F}$ to be intersecting. Define the set $\mathcal{I}(\mathcal{F}) = \{F\cap G: F, G \in \mathcal{F} \text{ and } F \neq G\}$ to be the collection of all distinct intersections of $\mathcal{F}$. Frankl et al. proved an upper bound for the size of $\mathcal{I}(\mathcal{F})$ of intersecting families $\mathcal{F}$ of $k$-subsets of $\{1,2,\ldots,n\}$. Their theorem holds for integers $n \geq 50 k^2$. In this article, we prove an upper bound for the size of $\mathcal{I}(\mathcal{F})$ of $t$-intersecting families $\mathcal{F}$, provided that $n$ exceeds a certain number $f(k,t)$. Along the way we also improve the threshold $k^2$ to $k^{3/2+o(1)}$ for the intersecting families.Comment: Some errors in the previous draft have been correcte

On rr-cross tt-intersecting families for vector spaces

Let $V$ be a vector space over a finite field $\mathbb{F}_q$ with dimension $n$, and ${V\brack k}$ the set of all subspaces of $V$ with dimension $k$. The families $\mathcal{F}_1,\dots,\mathcal{F}_r\subset{V\brack k}$ are called $r$-cross $t$-intersecting families if $\dim(F_1\cap F_2\cap\dots\cap F_r)\ge t$ for any $F_i\in\mathcal{F}_i$, $i=1,2,\dots,r$. In this paper, we prove a product version of the Hilton-Milner theorem for vector spaces, determining the structure of $2$-cross $t$-intersecting families $\mathcal{F}$, $\mathcal{G}$, with the maximum product of their sizes under the condition that both $\dim(\cap\{F: F\in\mathcal{F}\})$ and $\dim(\cap\{G: G\in\mathcal{G}\})$ are less than $t$. We also characterize the structure of $r$-cross $t$-intersecting families $\mathcal{F}_1$, $\mathcal{F}_2$, $\dots$, $\mathcal{F}_r$ with the maximum product of their sizes under the condition that $r\ge3$ and $\dim(\cap\{F: F\in\mathcal{F}_i\})<t$ for any $i\in\{1,2,\dots,r\}$