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arXiv · 1409.8665

Strong Turán stability

Mykhaylo Tyomkyn·Andrew J. Uzzell
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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 $ω(G)\leq r$ and $χ(G)\geq k$ for fixed $k \geq r \geq 2$.

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Mykhaylo Tyomkyn, Andrew J. Uzzell. 2014-09-30. Strong Turán stability. https://arxiv.org/abs/1409.8665

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