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

TL;DR

This paper refines the Erd ext{"o}s-Simonovits stability theorem by establishing an asymptotically sharp bound relating the parameters nd psilon, improving understanding of near-extremal K_{r+1}-free graphs.

Contribution

It provides an asymptotically optimal bound linking nd psilon in the stability theorem for K_{r+1}-free graphs, enhancing previous results.

Findings

Derived an asymptotically sharp bound between nd psilon.

Improved the quantitative understanding of the stability theorem.

Confirmed the bound's optimality as pproaches zero.

Abstract

The Erd\H{o}s-Simonovits stability theorem states that for all \epsilon >0 there exists \alpha >0 such that if G is a K_{r+1}-free graph on n vertices with e(G) > ex(n,K_{r+1}) - \alpha n^2, then one can remove \epsilon n^2 edges from G to obtain an r-partite graph. F\"uredi gave a short proof that one can choose \alpha=\epsilon. We give a bound for the relationship of \alpha and \varepsilon which is asymptotically sharp as \epsilon \to 0.