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Making Kr+1-free graphs r-partite

Published online by Cambridge University Press:  10 November 2020

József Balogh
Department of Mathematics, University of Illinois at Urbana-Champaign, Urbana, IL61801, USA, and Moscow Institute of Physics and Technology, Russian Federation
Felix Christian Clemen
Department of Mathematics, University of Illinois at Urbana-Champaign, Urbana, IL61801, USA
Mikhail Lavrov
Department of Mathematics, University of Illinois at Urbana-Champaign, Urbana, IL61801, USA
Bernard Lidický*
Department of Mathematics, Iowa State University, Ames, IA50011, USA
Florian Pfender
Department of Mathematical and Statistical Sciences, University of Colorado Denver, CO 80217-3364, USA
*Corresponding author. Email:
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The Erdős–Simonovits stability theorem states that for all ε > 0 there exists α > 0 such that if G is a Kr+1-free graph on n vertices with e(G) > ex(n, Kr+1)– α n2, then one can remove εn2 edges from G to obtain an r-partite graph. Füredi gave a short proof that one can choose α = ε. We give a bound for the relationship of α and ε which is asymptotically sharp as ε → 0.

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This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (, which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
© The Author(s), 2020. Published by Cambridge University Press


Research is partially supported by NSF grant DMS-1764123, Arnold O. Beckman Research Award (UIUC Campus Research Board RB 18132) and the Langan Scholar Fund (UIUC).

Research of this author is partially supported by NSF grant DMS-1855653.


Research of this author is partially supported by NSF grant DMS-1855622.


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