A note on $G$-intersecting families

TL;DR

This paper extends the known bounds for $G$-intersecting hypergraphs, showing that the Erdős–Ko–Rado type results hold for larger hypergraphs when the parameter $k$ is up to on the order of the square root of $n$, broadening the applicability of the theory.

Contribution

It generalizes previous results by extending the size range of hypergraphs for which the Erdős–Ko–Rado type theorems apply in the context of $G$-intersecting families.

Findings

Extended the bound to $k = O(\, ext{sqrt}(n))$ for $G$-intersecting hypergraphs.

Generalized Erdős–Ko–Rado theorem for sparse graphs.

Broadened the understanding of intersection properties in hypergraphs.

Abstract

Consider a graph $G$ and a $k$-uniform hypergraph $\mathcal{H}$ on common vertex set $[n]$. We say that $\mathcal{H}$ is $G$-intersecting if for every pair of edges in $X,Y \in \mathcal{H}$ there are vertices $x \in X$ and $y \in Y$ such that $x = y$ or $x$ and $y$ are joined by an edge in $G$. This notion was introduced by Bohman, Frieze, Ruszink\'o and Thoma who proved a natural generalization of the Erd\H{o}s-Ko-Rado Theorem for $G$-intersecting $k$-uniform hypergraphs for $G$ sparse and $k = O( n^{1/4} )$. In this note, we extend this result to $k = O\left( \sqrt{n} \right)$.