### Abstract

Let T_{n,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 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})-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 language | English |
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Pages (from-to) | 66-71 |

Number of pages | 6 |

Journal | Journal of Combinatorial Theory. Series B |

Volume | 115 |

DOIs | |

Publication status | Published - Nov 1 2015 |

### Fingerprint

### Keywords

- Extremal graphs
- Stability
- Turán number

### ASJC Scopus subject areas

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

### Cite this

**A proof of the stability of extremal graphs, Simonovits' stability from Szemerédi's regularity.** / Füredi, Z.

Research output: Contribution to journal › Article

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TY - JOUR

T1 - A proof of the stability of extremal graphs, Simonovits' stability from Szemerédi's regularity

AU - Füredi, Z.

PY - 2015/11/1

Y1 - 2015/11/1

N2 - 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].

AB - 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].

KW - Extremal graphs

KW - Stability

KW - Turán number

UR - http://www.scopus.com/inward/record.url?scp=84939263845&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84939263845&partnerID=8YFLogxK

U2 - 10.1016/j.jctb.2015.05.001

DO - 10.1016/j.jctb.2015.05.001

M3 - Article

AN - SCOPUS:84939263845

VL - 115

SP - 66

EP - 71

JO - Journal of Combinatorial Theory. Series B

JF - Journal of Combinatorial Theory. Series B

SN - 0095-8956

ER -