Allen, Peter, Böttcher, Julia, Hladký, Jan and Piguet, Diana. (2013) Turánnical hypergraphs. Random Structures & Algorithms, Vol.42 (No.1). pp. 29-58. ISSN 1042-9832

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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 = induces a copy of Kr. Firstly, we examine what happens when this restriction set is replaced by = {X∈ : 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 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 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

Item Type:	Journal Article
Subjects:	Q Science > QA Mathematics
Divisions:	Faculty of Science > Mathematics
Journal or Publication Title:	Random Structures & Algorithms
Publisher:	John Wiley & Sons, Inc.
ISSN:	1042-9832
Date:	2013
Volume:	Vol.42
Number:	No.1
Page Range:	pp. 29-58
Identification Number:	10.1002/rsa.20399
Status:	Peer Reviewed
Publication Status:	Published
Access rights to Published version:	Restricted or Subscription Access
URI:	http://wrap.warwick.ac.uk/id/eprint/47164

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