A structural result for hypergraphs with many restricted edge colorings

For k-uniform hypergraphs F and H and an integer r ≥ 2, let cr,F (H) denote the number of r-colorings of the set of hyperedges of H with no monochromatic copy of F and let cr,F (n) = maxH∈Hn cr,F (H), where the maximum runs over the family Hn of all k-uniform hypergraphs on n vertices. Moreover, let ex(n, F ) be the usual Turán function, i.e., the maximum number of hyperedges of an n-vertex k-uniform hypergraph which contains no copy of F . In this paper, we consider the question for determining cr,F (n) for arbitrary fixed hypergraphs F and show cr,F (n) = r ex(n,F )+o(n) for r = 2, 3. Moreover, we obtain a structural result for r = 2, 3 and any H with cr,F (H) ≥ rex(|V (H)|,F ) under the assumption that a stability result for the k-uniform hypergraph F exists and |V (H)| is sufficiently large. We also obtain exact results for cr,F (n) when F is a 3or 4-uniform generalized triangle and r = 2, 3, while cr,F (n) rex(n,F ) for r ≥ 4 and n sufficiently large.