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\(K^ s\)-free graphs without large \(K^ r\)-free subgraphs. (English) Zbl 0810.05040

For given natural numbers \(2\leq r< s\leq n\), the author improves an upper bound of Bollobás and Hind on the largest \(K_ r\)-free subgraph of an \(n\)-vertex \(K_ s\)-free graph. In case of \(r= 3\), \(s= 4\) the bound goes down from \(n^{.7+\epsilon}\) to \(n^{2/3}(\log n)^{1/3}\). He also improves the lower bound by a logarithmic factor. The proof of the upper bound is probabilistic and involves the Lovász Local Lemma and the exponential correlation inequality of Janson, Łuczak and the reviewer.
Very recently the author has improved his upper bound even further (to \(n^{3/5}(\log n)^{1/2}\) in case of \(r= 3\), \(s= 4\)). This new result will appear in Random Structures and Algorithms.

MSC:

05C35 Extremal problems in graph theory
05C80 Random graphs (graph-theoretic aspects)
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References:

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