Strong Tur\'an stability

TL;DR

This paper investigates the structure of nearly extremal $K_{r+1}$-free graphs, revealing they exhibit significant symmetry with most vertices having twins, and provides a new proof of a classical extremal graph theorem.

Contribution

It introduces a novel stability analysis showing that almost extremal graphs have high symmetry, simplifying the understanding of their structure.

Findings

Almost extremal $K_{r+1}$-free graphs have many twin vertices.

A new, concise proof of Simonovits' theorem on extremal graph structure.

Most vertices in such graphs exhibit a high degree of symmetry.

Abstract

We study the behaviour of $K_{r+1}$-free graphs $G$ of almost extremal size, that is, typically, $e(G)=ex(n,K_{r+1})-O(n)$. We show that such graphs must have a large amount of 'symmetry', in particular that all but very few vertices of $G$ must have twins. As a corollary, we obtain a new, short proof of a theorem of Simonovits on the structure of extremal graphs with $\omega(G)\leq r$ and $\chi(G)\geq k$ for fixed $k \geq r \geq 2$.