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On Colourings of Hypergraphs Without Monochromatic Fano Planes


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 , 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 = K and r = 2, Erdős and Rothschild conjectured that c2,K(n) = 2ex(n,K). 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.

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Combinatorics, Probability and Computing
  • ISSN: 0963-5483
  • EISSN: 1469-2163
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