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

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

Original languageEnglish
Pages (from-to)66-71
Number of pages6
JournalJournal of Combinatorial Theory. Series B
Publication statusPublished - Nov 1 2015



  • Extremal graphs
  • Stability
  • Turán number

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Discrete Mathematics and Combinatorics
  • Computational Theory and Mathematics

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