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Turánnical hypergraphs

Authors

  • Peter Allen,

    1. Instituto de Matemática e Estatística, Universidade de São Paulo, Rua do Matão 1010, São Paulo 05508–090, Brazil
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    • Supported 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 São Paulo.

  • Julia Böttcher,

    1. Instituto de Matemática e Estatística, Universidade de São Paulo, Rua do Matão 1010, São Paulo 05508–090, Brazil
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    • Supported by FAPESP (Proc. 2009/17831-7), and NUMEC/USP, Núcleo de Modelagem Estocástica e Complexidade of the University of São Paulo.

  • Jan Hladký,

    Corresponding author
    1. DIMAP and Mathematics Institute, University of Warwick, Coventry CV4 7AL, UK
    • DIMAP and Mathematics Institute, University of Warwick, Coventry CV4 7AL, UK
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    • Supported by DIMAP (EPSRC award EP/D063191/1).

  • Diana Piguet

    1. School of Mathematics, University of Birmingham, Edgbaston, Birmingham B15 2TT, UK
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    • Supported by DIMAP (EPSRC award EP/D063191/1).


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 equation image = equation image induces a copy of Kr.

Firstly, we examine what happens when this restriction set is replaced by equation image = {Xequation image: 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 equation image 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 equation image 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-Łuczak-Rödl Conjecture on Turán's theorem in random graphs.© 2012 Wiley Periodicals, Inc. Random Struct. Alg., 2012

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