TY - JOUR

T1 - Hypergraphs with Zero Chromatic Threshold

AU - Balogh, József

AU - Lenz, John

N1 - Funding 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.
Publisher Copyright:
© 2015, Springer Japan.

PY - 2016/7/1

Y1 - 2016/7/1

N2 - 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(|V(H)|r-1) 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.

AB - 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(|V(H)|r-1) 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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U2 - 10.1007/s00373-015-1667-6

DO - 10.1007/s00373-015-1667-6

M3 - Article

AN - SCOPUS:84949786291

SN - 0911-0119

VL - 32

SP - 1249

EP - 1262

JO - Graphs and Combinatorics

JF - Graphs and Combinatorics

IS - 4

ER -