Local Density In Graphs With Forbidden Subgraphs


A celebrated theorem of Turán asserts that every graph on n vertices with more than r− 1 2r n 2 edges contains a copy of a complete graph Kr+1. In this paper we consider the following more general question. Let G be a Kr+1-free graph of order n and let α be a constant, 0 < α 1. How dense can every induced subgraph of G on αn vertices be? We prove the following local density extension of Turán’s theorem. For every integer r 2 there exists a constant cr < 1 such that, if cr α 1 and every αn vertices of G span more than r− 1 2r (2α− 1)n

DOI: 10.1017/S0963548302005539

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@article{Keevash2003LocalDI, title={Local Density In Graphs With Forbidden Subgraphs}, author={Peter Keevash and Benny Sudakov}, journal={Combinatorics, Probability & Computing}, year={2003}, volume={12}, pages={139-153} }