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Hypergraphs with Zero Chromatic Threshold

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Abstract

Let F be an r-uniform hypergraph. The chromatic threshold of the family of F-free, r-uniform hypergraphs is the infimum of all non-negative reals c such that the subfamily of F-free, r-uniform hypergraphs H with minimum degree at least \(c \left( {\begin{array}{c}|V(H)|\\ r-1\end{array}}\right) \) has bounded chromatic number. The study of chromatic thresholds of various graphs has a long history, beginning with the early work of Erdős and Simonovits. One interesting problem, first proposed by Łuczak and Thomassé and then solved by Allen, Böttcher, Griffiths, Kohayakawa and Morris, is the characterization of graphs having zero chromatic threshold, in particular the fact that there are graphs with non-zero Turán density that have zero chromatic threshold. Here, we make progress on this problem for r-uniform hypergraphs, showing that a large class of hypergraphs have zero chromatic threshold in addition to exhibiting a family of constructions showing another large class of hypergraphs have non-zero chromatic threshold. Our construction is based on a particular product of the Bollobás–Erdős graphs defined earlier by the authors.

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Acknowledgments

We would like to thank the referees for careful reading of the manuscript and for numerous comments which greatly improved the quality of the manuscript. Also, we thank for Adam Zs. Wagner for some useful comments.

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Authors and Affiliations

  1. Department of Mathematics, University of Illinois, 1409 W. Green Street, Urbana, IL, 61801, USA

    József Balogh

  2. Mathematical Institute, University of Szeged, Szeged, Hungary

    József Balogh

  3. Department of Mathematics, Statistics, and Computer Science, University of Illinois at Chicago, 851 S. Morgan Street, Chicago, IL, USA

    John Lenz

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  1. József Balogh
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  2. John Lenz
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Correspondence to József Balogh.

Additional information

J. Balogh’s research supported in part by Marie Curie Fellowship IIF-327763, NSF CAREER Grant DMS-0745185, UIUC Campus Research Board Grants 11067 and 13039 (Arnold O. Beckman Research Award), and OTKA Grant K76099. J. Lenz’s research partly supported by NSA Grant H98230-13-1-0224.

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Balogh, J., Lenz, J. Hypergraphs with Zero Chromatic Threshold. Graphs and Combinatorics 32, 1249–1262 (2016). https://doi.org/10.1007/s00373-015-1667-6

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  • DOI: https://doi.org/10.1007/s00373-015-1667-6

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