Uniform Turán density of cycles

Abstract

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