Helly Theorems for Generalized Tur\'an Problems

TL;DR

This paper introduces a general theorem connecting the maximum number of tree copies in an -free graph to classical Ture1n numbers, utilizing novel Helly theorems for trees.

Contribution

It establishes a new framework linking Helly theorems for trees with generalized Ture1n problems, providing bounds and new insights.

Findings

Either x(n,T,) is ig-O of x(n,)^k or ig-Omega of n^{k+1}

Develops new variants of Helly Theorem for trees

First application of Helly theorems in Ture1n problems

Abstract

Given a graph $T$ and a family of graphs $\mathcal{F}$, the generalized Tur\'an number $\mathrm{ex}(n,T,\mathcal{F})$ is the maximum number of copies of $T$ in an $n$-vertex $\mathcal{F}$-free graph. We prove a general theorem which states that for any tree $T$, any family $\mathcal{F}$, and any integer $k$, either $\mathrm{ex}(n,T,\mathcal{F})$ is at least $\Omega(n^{k+1})$ or at most $O(\mathrm{ex}(n,\mathcal{F})^{k})$, from which we derive a number of consequences. Our proofs rely on new variants of the classical Helly Theorem for trees which may be of independent interest. As far as we are aware, this is the first known application of Helly theorems for Tur\'an type problems.