Turánnical hypergraphs

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 R = ` [n] r ́ induces a copy of Kr. Firstly, we examine what happens when this restriction set is replaced by R = {X ∈ ` [n] r ́ : X ∩ [m] 6= ∅}. 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 R on vertex set [n] is called Turánnical (respectively εTuránnical), if for any graph G on [n] with more edges than the Turán number tr(n) (respectively (1 + ε)tr(n) ), no hyperedge of R 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 KohayakawaLuczak-Rödl Conjecture on Turán’s theorem in random graphs.