Krivelevich, Michael \(K^ s\)-free graphs without large \(K^ r\)-free subgraphs. (English) Zbl 0810.05040 Comb. Probab. Comput. 3, No. 3, 349-354 (1994). 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. Reviewer: A.Ruciński (Poznań) Cited in 14 Documents MSC: 05C35 Extremal problems in graph theory 05C80 Random graphs (graph-theoretic aspects) Keywords:upper bound; Lovász Local Lemma; exponential correlation inequality PDFBibTeX XMLCite \textit{M. Krivelevich}, Comb. Probab. Comput. 3, No. 3, 349--354 (1994; Zbl 0810.05040) Full Text: DOI References: [1] Erd?s, Publ. Math. Inst. Hungar. Acad. Sci. 6 pp 181– (1961) [2] DOI: 10.1016/0012-365X(91)90042-Z · Zbl 0767.05083 · doi:10.1016/0012-365X(91)90042-Z [3] Alon, The probabilistic method (1992) [4] DOI: 10.1007/BF02579451 · Zbl 0491.05038 · doi:10.1007/BF02579451 [5] DOI: 10.1016/0097-3165(91)90074-Q · Zbl 0728.05059 · doi:10.1016/0097-3165(91)90074-Q [6] DOI: 10.1016/0097-3165(89)90064-2 · Zbl 0682.05005 · doi:10.1016/0097-3165(89)90064-2 [7] Janson, Random Graphs ’87 pp 73– (1990) [8] Erd?s, Canad. J. Math. 14 pp 702– (1962) · Zbl 0194.25304 · doi:10.4153/CJM-1962-060-4 [9] Erd?s, Infinite and finite sets pp 609– (1975) [10] DOI: 10.1006/jctb.1994.1026 · Zbl 0809.05061 · doi:10.1006/jctb.1994.1026 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.