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- W3215696486 abstract "In the early 1980s, Erdős and Sos initiated the study of the classical Turan problem with a uniformity condition: the uniform Turan 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 Turan 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. 97 (2018), 77--97], while the latter still remains open for almost 40 years. In addition to $K_4^{(3)-}$, the only $3$-uniform hypergraphs whose uniform Turan density is known are those with zero uniform Turan density classified by Reiher, Rodl and Schacht [J. London Math. Soc. 97 (2018), 77-97] and a specific family with uniform Turan 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 Turan density of a fundamental family of $3$-uniform hypergraphs, namely tight cycles $C_ell^{(3)}$. The uniform Turan density of $C_ell^{(3)}$, $ellge 5$, 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." @default.
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- W3215696486 date "2021-12-02" @default.
- W3215696486 modified "2023-09-26" @default.
- W3215696486 title "Uniform Tur'an density of cycles" @default.
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