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Hypergraphs with uniform Turán density equal to 8/27

Part of: Extremal combinatorics Graph theory

Published online by Cambridge University Press:  17 December 2025

Frederik Garbe Open the ORCID record for Frederik Garbe [Opens in a new window] ,
Daniel Il’kovič ,
Daniel Kráľ Open the ORCID record for Daniel Kráľ [Opens in a new window] ,
Filip Kučerák  and
Ander Lamaison Open the ORCID record for Ander Lamaison [Opens in a new window]
Frederik Garbe*
Affiliation:
Czech Academy of Sciences, Institute of Computer Science, Prague, Czech Republic Faculty of Informatics, Masaryk University, Brno, Czech Republic
Daniel Il’kovič
Affiliation:
Faculty of Informatics, Masaryk University, Brno, Czech Republic Institute of Mathematics, Leipzig University, Leipzig, Germany Mathematics Institute and DIMAP, University of Warwick, Coventry, UK
Daniel Kráľ
Affiliation:
Faculty of Informatics, Masaryk University, Brno, Czech Republic Institute of Mathematics, Leipzig University, Leipzig, Germany Max Planck Institute for Mathematics in the Sciences, Leipzig, Germany
Filip Kučerák
Affiliation:
Faculty of Informatics, Masaryk University, Brno, Czech Republic Max Planck Institute for Mathematics in the Sciences, Leipzig, Germany
Ander Lamaison
Affiliation:
Faculty of Informatics, Masaryk University, Brno, Czech Republic Extremal Combinatorics and Probability Group (ECOPRO), Institute for Basic Science (IBS), Daejeon, South Korea
*
Corresponding author: Frederik Garbe; Email: garbe@cs.cas.cz
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Abstract

In the 1980s, Erdős and Sós initiated the study of Turán problems with a uniformity condition on the distribution of edges: 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 asked to determine the uniform Turán densities of $K_4^{(3)-}$ and $K_4^{(3)}$. After more than 30 years, the former was solved in [Israel J. Math. 211 (2016), 349 – 366] and [J. Eur. Math. Soc. 20 (2018), 1139 – 1159], while the latter still remains open. Till today, there are known constructions of $3$-uniform hypergraphs with uniform Turán density equal to $0$, $1/27$, $4/27$, and $1/4$ only. We extend this list by a fifth value: we prove an easy to verify sufficient condition for the uniform Turán density to be equal to $8/27$ and identify hypergraphs satisfying this condition.

Keywords

hypergraphsuniform Turán density

MSC classification

Primary: 05C65: Hypergraphs
Secondary: 05D99: None of the above, but in this section

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Type
Paper
Information
Combinatorics, Probability and Computing , First View , pp. 1 - 10
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Copyright
© The Author(s), 2025. Published by Cambridge University Press

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