Exact Results on the Number of Restricted Edge Colorings for Some Families of Linear Hypergraphs

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 is over the family Hn of all k-uniform hypergraphs on n vertices. Moreover, let ex(n, F ) be the usual extremal function, i.e., the maximum number of hyperedges of an n-vertex k-uniform hypergraph which contains no copy of F . Here, we consider the question for determining cr,F (n) for F being the kuniform expanded, complete 2-graph Hk `+1 or the k-uniform Fan(k)-hypergraph F k `+1 with core of size (` + 1), where ` ≥ k ≥ 3, and we show cr,F (n) = r ex(n,F ) for r = 2, 3 and n large enough. Moreover, for r = 2 or r = 3, for k-uniform hypergraphs H on n vertices the equality cr,F (H) = r ex(n,F ) only holds if H is isomorphic to the `-partite, k-uniform Turán hypergraph on n vertices, once n is large enough. On the other hand, we show that cr,F (n) is exponentially larger than rex(n,F ), if r ≥ 4.