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.

中文翻译：

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制作无 Kr+1 的图 r-partite

Erdős–Simonovits 稳定性定理指出，对于所有ε> 0 存在α> 0 使得如果G是一个ķr+1- 免费图表n顶点与e(G) > 前(n,ķr+1)–α n2，然后可以删除εn2边缘从G获得一个r- 分图。Füredi 给出了一个可以选择的简短证明α=ε. 我们给出了关系的界限α和ε这是渐近尖锐的ε→ 0。