Hypergraphs with minimum positive uniform Tur\'an density

TL;DR

This paper constructs 3-uniform hypergraphs with a uniform Turán density exactly equal to 1/27, addressing a question about the possible values of this density.

Contribution

The authors provide the first explicit construction of 3-uniform hypergraphs with Turán density exactly 1/27, demonstrating that this value is attainable.

Findings

Constructed hypergraphs with Turán density exactly 1/27

Confirmed the density value is achievable, answering a prior open question

Contributed to understanding the range of Turán densities for 3-uniform hypergraphs

Abstract

Reiher, R\"odl and Schacht [J. London Math. Soc. 97 (2018), 77--97] showed that the uniform Tur\'an density of every $3$-uniform hypergraph is either $0$ or at least $1/27$, and asked whether there exist $3$-uniform hypergraphs with uniform Tur\'an density equal or arbitrarily close to $1/27$. We construct $3$-uniform hypergraphs with uniform Tur\'an density equal to $1/27$.