Tur\'annical hypergraphs

TL;DR

This paper explores how replacing dense global restrictions in Turan's theorem with sparser or localized ones affects extremal graph properties, including thresholds for random hypergraphs and graphs to be Turannical.

Contribution

It introduces the concept of Turannical hypergraphs, analyzes restrictions on K_r copies touching specific sets, and determines thresholds for sparse random hypergraphs and graphs to be Turannical.

Findings

Replaced global restrictions with localized ones in Turan's theorem.

Determined thresholds for random hypergraphs to be Turannical.

Extended results to sparse random graphs using recent techniques.

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 Turan's theorem, which deals with graphs G=([n],E) such that no member of the restriction set consisting of all r-tuples on [n] induces a copy of K_r. Firstly, we examine what happens when this restriction set is replaced just by all r-tuples touching a given m-element set. That is, we determine the maximal number of edges in an n-vertex such that no K_r hits a given vertex set. Secondly, we consider sparse random restriction sets. An r-uniform hypergraph R on vertex set [n] is called Turannical (respectively epsilon-Turannical), if for any graph G on [n] with more edges than the Turan number ex(n,K_r) (respectively (1+\eps)ex(n,K_r), no hyperedge of R induces a copy of K_r in G. We determine the thresholds for random r-uniform hypergraphs to be Turannical and to epsilon-Turannical. 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-Luczak-Rodl Conjecture on Turan's theorem in random graphs.