# Hypergraphs with minimum positive uniform Turán density

## Frederik Garbe

*Masaryk University*

March 31, 2022, 12:20 in S6

## Abstract

Determining Turán densities in the setting of k-uniform hypergraphs is a challenging problem, even the Turán density of the complete 3-uniform 4-vertex hypergraph is not known. Since the edges in the conjectured extremal constructions for those problems are often distributed in a non-uniform way, already Erdős and Sós suggested to study the notion of a uniform Turán density of hypergraphs, which requires the edges of the host hypergraph to be distributed uniformly. Reiher, Rödl and Schacht showed that the uniform Turán density of every 3-uniform hypergraph is either 0 or at least 1/27, and asked whether there exist 3-uniform hypergraphs with uniform Turán density equal or arbitrarily close to 1/27. We construct 3-uniform hypergraphs with uniform Turán density equal to 1/27.

This is joint work with Daniel Král' and Ander Lamaison.