Turánnical hypergraphs |
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Authors: | Peter Allen Julia Böttcher Jan Hladký Diana Piguet |
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Institution: | 1. Instituto de Matemática e Estatística, Universidade de S?o Paulo, Rua do Mat?o 1010, S?o Paulo 05508–090, BrazilSupported by DIMAP (EPSRC award EP/D063191/1), FAPESP (Proc. 2010/09555‐7), and NUMEC/USP, Núcleo de Modelagem Estocástica e Complexidade of the University of S?o Paulo.;2. Instituto de Matemática e Estatística, Universidade de S?o Paulo, Rua do Mat?o 1010, S?o Paulo 05508–090, BrazilSupported by FAPESP (Proc. 2009/17831‐7), and NUMEC/USP, Núcleo de Modelagem Estocástica e Complexidade of the University of S?o Paulo.;3. DIMAP and Mathematics Institute, University of Warwick, Coventry CV4 7AL, UKSupported by DIMAP (EPSRC award EP/D063191/1).;4. School of Mathematics, University of Birmingham, Edgbaston, Birmingham B15 2TT, UKSupported by DIMAP (EPSRC award EP/D063191/1). |
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Abstract: | This paper is motivated by the question of how global and dense restriction sets in results from extremal combinatorics can be replaced by less global and sparser ones. The result we consider here as an example is Turán's theorem, which deals with graphs G = (n],E) such that no member of the restriction set \begin{align*}\mathcal {R}\end{align*} = \begin{align*}\left( {\begin{array}{*{20}c} {n]} \\ r \\ \end{array} } \right)\end{align*} induces a copy of Kr. Firstly, we examine what happens when this restriction set is replaced by \begin{align*}\mathcal {R}\end{align*} = {X∈ \begin{align*}\left( {\begin{array}{*{20}c} {n]} \\ r \\ \end{array} } \right)\end{align*}: X ∩ m]≠??}. That is, we determine the maximal number of edges in an n ‐vertex such that no Kr hits a given vertex set. Secondly, we consider sparse random restriction sets. An r ‐uniform hypergraph \begin{align*}\mathcal R\end{align*} on vertex set n] is called Turánnical (respectively ε ‐Turánnical), if for any graph G on n] with more edges than the Turán number tr(n) (respectively (1 + ε)tr(n) ), no hyperedge of \begin{align*}\mathcal {R}\end{align*} induces a copy of Kr in G. We determine the thresholds for random r ‐uniform hypergraphs to be Turánnical and to be ε ‐Turánnical. Thirdly, we transfer this result to sparse random graphs, using techniques recently developed by Schacht Extremal results for random discrete structures] to prove the Kohayakawa‐?uczak‐Rödl Conjecture on Turán's theorem in random graphs.© 2012 Wiley Periodicals, Inc. Random Struct. Alg., 2012 |
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Keywords: | Turá n's theorem extremal combinatorics random hypergraphs |
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