TY - JOUR

T1 - UNIFORM TURÁN DENSITY OF CYCLES

AU - Bucić, Matija

AU - Cooper, Jacob W.

AU - Král, Daniel

AU - Mohr, Samuel

AU - Correia, David Munhá

N1 - Publisher Copyright:
© 2023 American Mathematical Society.

PY - 2023/7

Y1 - 2023/7

N2 - In the early 1980s, Erdos and Śos initiated the study of the classical Turán problem with a uniformity condition: the uniform Turán 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 subhypergraphs have density at least d contains H. In particular, they raise the questions of determining the uniform Turán densities of K(3)- 4 and K(3) 4 . 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(3)- 4 , the only 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. 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-uniform hypergraphs, namely tight cycles C(3) ℓ . The uniform Turán density of C(3) ℓ , ℓ ≥ 5, is equal to 4/27 if ℓ is not divisible by three, and is equal to zero otherwise. The case ℓ = 5 resolves a problem suggested by Reiher.

AB - In the early 1980s, Erdos and Śos initiated the study of the classical Turán problem with a uniformity condition: the uniform Turán 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 subhypergraphs have density at least d contains H. In particular, they raise the questions of determining the uniform Turán densities of K(3)- 4 and K(3) 4 . 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(3)- 4 , the only 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. 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-uniform hypergraphs, namely tight cycles C(3) ℓ . The uniform Turán density of C(3) ℓ , ℓ ≥ 5, is equal to 4/27 if ℓ is not divisible by three, and is equal to zero otherwise. The case ℓ = 5 resolves a problem suggested by Reiher.

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U2 - 10.1090/tran/8873

DO - 10.1090/tran/8873

M3 - Article

AN - SCOPUS:85160944208

SN - 0002-9947

VL - 376

SP - 4765

EP - 4809

JO - Transactions of the American Mathematical Society

JF - Transactions of the American Mathematical Society

IS - 7

ER -