Every graph is eventually Tur\'an-good

TL;DR

This paper proves that for sufficiently large r, the r-partite Turán graph maximizes the number of copies of a given graph H among all K_{r+1}-free graphs, confirming a conjecture.

Contribution

It establishes that the Turán graph is asymptotically optimal for counting copies of H in K_{r+1}-free graphs for large r, confirming a conjecture by Gerbner and Palmer.

Findings

Turán graph maximizes copies of H for large r

Confirms a conjecture by Gerbner and Palmer

Provides asymptotic extremal results for graph copies

Abstract

Let $H$ be a graph. We show that if $r$ is large enough as a function of $H$, then the $r$-partite Tur\'an graph maximizes the number of copies of $H$ among all $K_{r+1}$-free graphs on a given number of vertices. This confirms a conjecture of Gerbner and Palmer.