# $H$-free subgraphs of dense graphs maximizing the number of cliques and their blow-ups

## TL;DR

This paper investigates the structure of dense, $H$-free subgraphs that maximize the number of cliques and their blow-ups, revealing their colorability and extending to various forbidden graphs and configurations.

## Contribution

It establishes that such extremal subgraphs are $(k-1)$-colorable under certain conditions and extends results to general $H$ and balanced blowups.

## Key findings

- Maximal $H$-free subgraphs are $(k-1)$-colorable in dense graphs.
- Results apply to graphs with high minimum degree and specific chromatic properties.
- Extensions cover general forbidden graphs and maximization of balanced blowups.

## Abstract

We consider the structure of $H$-free subgraphs of graphs with high minimal degree. We prove that for every $k>m$ there exists an $\epsilon:=\epsilon(k,m)>0$ so that the following holds. For every graph $H$ with chromatic number $k$ from which one can delete an edge and reduce the chromatic number, and for every graph $G$ on $n>n_0(H)$ vertices in which all degrees are at least $(1-\epsilon)n$, any subgraph of $G$ which is $H$-free and contains the maximum number of copies of the complete graph $K_m$ is $(k-1)$-colorable. We also consider several extensions for the case of a general forbidden graph $H$ of a given chromatic number, and for subgraphs maximizing the number of copies of balanced blowups of complete graphs.