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 $c_{r,F}(n)=\max_{H\in\ccHn} c_{r,F}(H)$ , where the maximum runs over all k-uniform hypergraphs on n vertices. Moreover, let ex(n,F) be the usual extremal or Turan 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.