Published online by Cambridge University Press: 10 November 2020
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.
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.