Strong Turán Stability

  • Mykhaylo Tyomkyn
  • Andrew J. Uzzell
Keywords: Forbidden subgraph, stability, saturation


We study maximal $K_{r+1}$-free graphs $G$ of almost extremal size—typically, $e(G)=\operatorname{ex}(n,K_{r+1})-O(n)$. We show that any such graph $G$ must have a large amount of `symmetry': in particular, all but very few vertices of $G$ must have twins. (Two vertices $u$ and $v$ are twins if they have the same neighbourhood.) As a corollary, we obtain a new, short proof of a theorem of Simonovits on the structure of extremal $K_{r+1}$-free graphs of chromatic number at least $k$ for all fixed $k \geq r \geq 2$.
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