Making $K_{r+1}$-Free Graphs $r$-partite

Abstract

The Erdős-Simonovits stability theorem states that for all ε > 0 there exists α > 0 such that if G is a K r+1 -free graph on n vertices with e(G) > ex(n, K r+1 ) − αn 2 , then one can remove εn 2 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.

Large independent sets from local considerations

“…So we are done unless t < 3 2 m. Since G is K r -free and has t r−1 (m) − t edges a stability theorem (see Theorem 1.3 in [6]) implies that G can be made r − 1-partite by removing at most rt 3/2 2m edges (being crude and using that m is sufficiently larger than r). Translating this to G we conclude G is a vertex disjoint union of r − 1 cliques missing a few edges, in total at most rt 3/2 2m ≤ r √ m edges.…”

“…Remark. The stability result we used above was also independently discovered in [33] (in an asymptotic form), we used the variant from [6] since it is explicit. Our problem seems to be closely related to this type of stability problems for Turán's theorem.…”