Density of small diameter subgraphs in $K_r$-free graphs

TL;DR

This paper investigates the maximum number of small subgraphs in $K_r$-free graphs and disproves a recent conjecture about their asymptotic structure.

Contribution

It provides a counterexample to a conjecture on the extremal structure of subgraph counts in $K_r$-free graphs.

Findings

Disproves the conjecture by Grzesik et al.

Shows that the maximum is not always attained by blow-ups of $K_{r-1}$.

Clarifies conditions for subgraph density in $K_r$-free graphs.

Abstract

We denote by $\text{ex}(n, H, F)$ the maximum number of copies of $H$ in an $n$-vertex graph that does not contain $F$ as a subgraph. Recently, Grzesik, Gy\H{o}ri, Salia, Tompkins considered conditions on $H$ under which $\text{ex}(n, H, K_r)$ is asymptotically attained at a blow-up of $K_{r-1}$, and proposed a conjecture. In this note we disprove their conjecture.