A proof of the stability of extremal graphs, Simonovits' stability from Szemerédi's regularity

Let T n , p denote the complete p-partite graph of order n having the maximum number of edges. The following sharpening of Turan's theorem is proved. Every K p + 1 -free graph with n vertices and e ( T n , p ) - t edges contains a p-partite subgraph with at least e ( T n , p ) - 2 t edges.As a corollary of this result we present a concise, contemporary proof (i.e., one applying the Removal Lemma, a corollary of Szemeredi's regularity lemma) for the classical stability result of Simonovits 25.

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