The density Tur\'an problem for hypergraphs

TL;DR

This paper investigates the minimal density conditions in hypergraph blow-ups that guarantee the presence of a transversal copy of a subgraph, extending known graph results to hypergraphs using entropy compression techniques.

Contribution

The paper introduces new upper bounds for the hypergraph density Turán problem, generalizing previous graph results with novel entropy compression methods.

Findings

Established upper bounds for hypergraph density Turán problem

Extended graph bounds to hypergraphs

Applied entropy compression in combinatorial proofs

Abstract

Given a $k$-graph $H$ a complete blow-up of $H$ is a $k$-graph $\hat{H}$ formed by replacing each $v\in V(H)$ by a non-empty vertex class $A_v$ and then inserting all edges between any $k$ vertex classes corresponding to an edge of $H$. Given a subgraph $G\subseteq \hat{H}$ and an edge $e\in E(H)$ we define the density $d_e(G)$ to be the proportion of edges present in $G$ between the classes corresponding to $e$. The density Tur\'an problem for $H$ asks: determine the minimal value $d_{crit}(H)$ such that any subgraph $G\subseteq \hat{H}$ satisfying $d_e(G)> d_{crit}(H)$ for every $e\in E(H)$ contains a copy of $H$ as a transversal, i.e. a copy of $H$ meeting each vertex class of $\hat{H}$ exactly once. We give upper bounds for this hypergraph density Tur\'an problem that generalise the known bounds for the case of graphs due to Csikv\'ari and Nagy, [Combinatorics, Probability and Computing, 21(4):531-553, 2012] although our methods are different, employing an entropy compression argument.