@article{BCKMM23uniformturandensitycycles,
title={Uniform Tur{\'a}n density of cycles},
author={Buci{\'c}, Matija and Cooper, Jacob W and Kr{\'a}{\v{l}}, Daniel and Mohr, Samuel and Munh{\'a}, David Correia},
note = {submitted. },
archivePrefix = {arXiv},
eprint={2112.01385},
biburl = {http://samuel-mohr.de/files/bib/21.bib},
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 \href{https://doi.org/10.1007/s11856-015-1267-4}{[Israel J. Math. 211 (2016), 349--366]} and \href{https://doi.org/10.4171/JEMS/784}{[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 \href{https://doi.org/10.1112/jlms.12095}{[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.}
}