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 - Funding Information:
The work of the third and fourth authors had received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No 648509). This publication reflects only its authors’ view; the European Research Council Executive Agency is not responsible for any use that may be made of the information it contains. The second, third and fourth authors were also supported by the MUNI Award in Science and Humanities of the Grant Agency of Masaryk University. The work of the fifth author was supported in part by the SNSF grant 200021 196965.
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 -