Elsevier

Discrete Mathematics

Volume 339, Issue 12, 6 December 2016, Pages 3020-3031
Discrete Mathematics

Colorings of hypergraphs with large number of colors

https://doi.org/10.1016/j.disc.2016.06.016Get rights and content
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Abstract

The paper deals with the well-known problem of Erdős and Hajnal concerning colorings of uniform hypergraphs and some related questions. Let m(n,r) denote the minimum possible number of edges in an n-uniform non-r-colorable hypergraph. We show that for r>n, c1nlnnm(n,r)rnC1n3lnn, where c1,C1>0 are some absolute constants. Moreover, we obtain similar bounds for d(n,r), which is equal to the minimum possible value of the maximum edge degree in an n-uniform non-r-colorable hypergraph. If r>n, then c2nlnnd(n,r)rn1C2n3lnn, where c2,C2>0 are some other absolute constants.

Keywords

Colorings of hypergraphs
Property B problem
Turán numbers

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