A Stability Theorem for Maximal $K_{r+1}$-free Graphs

TL;DR

This paper proves a stability theorem for maximal $K_{r+1}$-free graphs, showing they contain large complete $r$-partite subgraphs under certain edge conditions, and establishes the optimality of this result.

Contribution

It provides a new stability result for maximal $K_{r+1}$-free graphs, answering a question posed by Tyomkyn and Uzzell.

Findings

Maximal $K_{r+1}$-free graphs with nearly extremal edges contain large complete $r$-partite subgraphs.

The stability result is shown to be optimal.

The paper resolves an open question in extremal graph theory.

Abstract

For $r \geq 2$, we show that every maximal $K_{r+1}$-free graph $G$ on $n$ vertices with $(1-\frac{1}{r})\frac{n^2}{2}-o(n^{\frac{r+1}{r}})$ edges contains a complete $r$-partite subgraph on $(1 - o(1))n$ vertices. We also show that this is best possible. This result answers a question of Tyomkyn and Uzzell.