Uniform Tur\'an density of cycles

TL;DR

This paper determines the uniform Turán density of tight cycles in 3-uniform hypergraphs, revealing a pattern based on cycle length divisibility, and introduces new embedding tools for hypergraph density problems.

Contribution

It completely characterizes the uniform Turán density of all tight cycles in 3-uniform hypergraphs, solving a long-standing open problem for cycles of length 5.

Findings

Uniform Turán density of $C_ ext{ell}^{(3)}$ is $4/27$ for $ ext{ell} ot\equiv 0 mod 3$.

Uniform Turán density of $C_ ext{ell}^{(3)}$ is zero when $ ext{ell}$ is divisible by 3.

New embedding tools for hypergraphs with positive density are developed.

Abstract

In the early 1980s, Erd\H{o}s and S\'os initiated the study of the classical Tur\'an problem with a uniformity condition: the uniform Tur\'an density of a hypergraph $H$ is the infimum over all $d$ for which any sufficiently large hypergraph with the property that all its linear-size subhyperghraphs have density at least $d$ contains $H$. In particular, they raise the questions of determining the uniform Tur\'an densities of $K_4^{(3)-}$ and $K_4^{(3)}$. The former question was solved only recently in [Israel J. Math. 211 (2016), 349-366] and [J. Eur. Math. Soc. 20 (2018), 1139-1159], while the latter still remains open for almost 40 years. In addition to $K_4^{(3)-}$, the only $3$-uniform hypergraphs whose uniform Tur\'an density is known are those with zero uniform Tur\'an density classified by Reiher, R\"odl and Schacht [J. London Math. Soc. 97 (2018), 77-97] and a specific family with uniform Tur\'an density equal to $1/27$. We develop new tools for embedding hypergraphs in host hypergraphs with positive uniform density and apply them to completely determine the uniform Tur\'an density of a fundamental family of $3$-uniform hypergraphs, namely tight cycles $C_\ell^{(3)}$. The uniform Tur\'an density of $C_\ell^{(3)}$, $\ell\ge 5$, is equal to $4/27$ if $\ell$ is not divisible by three, and is equal to zero otherwise. The case $\ell=5$ resolves a problem suggested by Reiher.