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  • Cited by 6
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    This article has been cited by the following publications. This list is generated based on data provided by CrossRef.

    Ceyhan, Elvan 2016. Edge density of new graph types based on a random digraph family. Statistical Methodology,

    Bollobás, Béla and Scott, Alex 2015. Intersections of hypergraphs. Journal of Combinatorial Theory, Series B, Vol. 110, p. 180.

    Bollobás, Béla and Scott, Alex 2011. Intersections of graphs. Journal of Graph Theory, Vol. 66, Issue. 4, p. 261.

    Sudakov, Benny 2007. Note making a K 4-free graph bipartite. Combinatorica, Vol. 27, Issue. 4, p. 509.

    Keevash, Peter Mubayi, Dhruv and Wilson, Richard M. 2006. Set Systems with No Singleton Intersection. SIAM Journal on Discrete Mathematics, Vol. 20, Issue. 4, p. 1031.

    Keevash, Peter and Sudakov, Benny 2006. Sparse halves in triangle-free graphs. Journal of Combinatorial Theory, Series B, Vol. 96, Issue. 4, p. 614.

  • Combinatorics, Probability and Computing, Volume 12, Issue 2
  • March 2003, pp. 139-153

Local Density in Graphs with Forbidden Subgraphs

  • DOI:
  • Published online: 01 March 2003

A celebrated theorem of Turán asserts that every graph on n vertices with more than $\frac{r\,{-}\,1}{2r}n^2$ edges contains a copy of a complete graph $K_r+1$. In this paper we consider the following more general question. Let G be a $K_r+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 $r\geq 2$ there exists a constant $c_r < 1$ such that, if $c_r < \alpha < 1$ and every αn vertices of G span more than

\[ \frac{r\,{-}\,1}{2r}(2\alpha -1)n^2\vspace*{7pt} \]

edges, then G contains a copy of $K_r+1$. This result is clearly best possible and answers a question of Erdős, Faudree, Rousseau and Schelp [5].

In addition, we prove that the only $K_r+1-free graph of order n, in which every αn vertices span at least $\frac{r\,{-}\,1}{2r}(2\alpha -1)n^2$ edges, is a Turán graph. We also obtain the local density version of the Erdős–Stone theorem.

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Combinatorics, Probability and Computing
  • ISSN: 0963-5483
  • EISSN: 1469-2163
  • URL: /core/journals/combinatorics-probability-and-computing
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