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On Colourings of Hypergraphs Without Monochromatic Fano Planes

Published online by Cambridge University Press:  01 September 2009

HANNO LEFMANN ,
YURY PERSON ,
VOJTĚCH RÖDL  and
MATHIAS SCHACHT
HANNO LEFMANN
Affiliation:
Fakultät für Informatik, Technische Universität Chemnitz, D-09107 Chemnitz, Germany (e-mail: lefmann@informatik.tu-chemnitz.de)
YURY PERSON
Affiliation:
Institut für Informatik, Humboldt-Universität zu Berlin, D-10099 Berlin, Germany (e-mail: person@informatik.hu-berlin.de, schacht@informatik.hu-berlin.de)
VOJTĚCH RÖDL
Affiliation:
Department of Mathematics and Computer Science, Emory University, Atlanta, GA 30322, USA (e-mail: rodl@mathcs.emory.edu)
MATHIAS SCHACHT
Affiliation:
Institut für Informatik, Humboldt-Universität zu Berlin, D-10099 Berlin, Germany (e-mail: person@informatik.hu-berlin.de, schacht@informatik.hu-berlin.de)
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Abstract

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For k-uniform hypergraphs F and H and an integer r, let cr,F(H) denote the number of r-colourings of the set of hyperedges of H with no monochromatic copy of F, and let $c_{r,F}(n)=\max_{H\in\ccHn} c_{r,F}(H)$, where the maximum runs over all k-uniform hypergraphs on n vertices. Moreover, let ex(n,F) be the usual extremal or Turán function, i.e., the maximum number of hyperedges of an n-vertex k-uniform hypergraph which contains no copy of F.

For complete graphs F = Kℓ and r = 2, Erdős and Rothschild conjectured that c2,Kℓ(n) = 2ex(n,Kℓ). This conjecture was proved by Yuster for ℓ = 3 and by Alon, Balogh, Keevash and Sudakov for arbitrary ℓ. In this paper, we consider the question for hypergraphs and show that, in the special case when F is the Fano plane and r = 2 or 3, then cr,F(n) = rex(n,F), while cr,F(n) ≫ rex(n,F) for r ≥ 4.

Information

Type
Paper
Information
Combinatorics, Probability and Computing , Volume 18 , Issue 5: New Directions in Algorithms, Combinatorics and Optimization , September 2009 , pp. 803 - 818
Copyright
Copyright © Cambridge University Press 2009

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