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  • Cited by 9
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    This chapter has been cited by the following publications. This list is generated based on data provided by CrossRef.

    Wu, Biao and Peng, Yuejian 2018. Dense 3-uniform hypergraphs containing a large clique. Science China Mathematics, Vol. 61, Issue. 3, p. 577.

    Keller, Chaya Smorodinsky, Shakhar and Tardos, Gábor 2018. Improved bounds on the Hadwiger–Debrunner numbers. Israel Journal of Mathematics, Vol. 225, Issue. 2, p. 925.

    Linial, Nati Newman, Ilan Peled, Yuval and Rabinovich, Yuri 2018. Extremal hypercuts and shadows of simplicial complexes. Israel Journal of Mathematics,

    BUKH, BORIS and JIANG, ZILIN 2017. A Bound on the Number of Edges in Graphs Without an Even Cycle. Combinatorics, Probability and Computing, Vol. 26, Issue. 01, p. 1.

    BALOGH, JÓZSEF BUTTERFIELD, JANE HU, PING LENZ, JOHN and MUBAYI, DHRUV 2016. On the Chromatic Thresholds of Hypergraphs. Combinatorics, Probability and Computing, Vol. 25, Issue. 02, p. 172.

    Janson, Svante and Warnke, Lutz 2016. The lower tail: Poisson approximation revisited. Random Structures & Algorithms, Vol. 48, Issue. 2, p. 219.

    Pikhurko, Oleg 2014. ON possible Turán densities. Israel Journal of Mathematics, Vol. 201, Issue. 1, p. 415.

    Hoppen, Carlos Kohayakawa, Yoshiharu and Lefmann, Hanno 2013. Information Theory, Combinatorics, and Search Theory. Vol. 7777, Issue. , p. 432.

    Pikhurko, Oleg 2011. The minimum size of 3-graphs without a 4-set spanning no or exactly three edges. European Journal of Combinatorics, Vol. 32, Issue. 7, p. 1142.

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  • Print publication year: 2011
  • Online publication date: August 2011

3 - Hypergraph Turán problems

Summary

Abstract

One of the earliest results in Combinatorics is Mantel's theorem from 1907 that the largest triangle-free graph on a given vertex set is complete bipartite. However, a seemingly similar question posed by Turán in 1941 is still open: what is the largest 3-uniform hypergraph on a given vertex set with no tetrahedron? This question can be considered a test case for the general hypergraph Turán problem, where given an r-uniform hypergraph F, we want to determine the maximum number of edges in an r-uniform hypergraph on n vertices that does not contain a copy of F. To date there are very few results on this problem, even asymptotically. However, recent years have seen a revitalisation of this field, via significant developments in the available methods, notably the use of stability (approximate structure) and flag algebras. This article surveys the known results and methods, and discusses some open problems.

Acknowledgements

Research supported in part by ERC grant 239696 and EPSRC grant EP/G056730/1. Thanks to Dan Hefetz, Dhruv Mubayi, Richard Mycroft and Oleg Pikhurko for helpful comments and corrections.

Introduction

The Turán number ex(n, F) is the maximum number of edges in an F-free r-graph on n vertices. It is a long-standing open problem in Extremal Combinatorics to develop some understanding of these numbers for general r-graphs F. Ideally, one would like to compute them exactly, but even asymptotic results are currently only known in certain cases. For ordinary graphs (r = 2) the picture is fairly complete.

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Surveys in Combinatorics 2011
  • Online ISBN: 9781139004114
  • Book DOI: https://doi.org/10.1017/CBO9781139004114
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