Chromatic Ramsey number of acyclic hypergraphs

Suppose that T is an acyclic r-uniform hypergraph, with r ≥ 2. We define the (t-color) chromatic Ramsey number χ(T, t) as the smallest m with the following property: if the edges of any m-chromatic r-uniform hypergraph are colored with t colors in any manner, there is a monochromatic copy of T . We observe that χ(T, t) is well defined and ⌈ R(T, t)− 1 r − 1 ⌉ + 1 ≤ χ(T, t) ≤ |E(T )| + 1 where R(T, t) is the t-color Ramsey number of H. We give linear upper bounds for χ(T, t) when T is a matching or star, proving that for r ≥ 2, k ≥ 1, t ≥ 1, χ(M k , t) ≤ (t− 1)(k − 1) + 2k and χ(S k, t) ≤ t(k − 1) + 2 where M k and S r k are, respectively, the r-uniform matching and star with k edges. The general bounds are improved for 3-uniform hypergraphs. We prove that χ(M k , 2) = 2k, extending a special case of Alon-Frankl-Lovász’ theorem. We also prove that χ(S 2 , t) ≤ t+1, which is sharp for t = 2, 3. This is a corollary of a more general result. We define H [1] as the 1-intersection graph of H, whose vertices represent hyperedges and whose edges represent intersections of hyperedges in exactly one vertex. We prove that χ(H) ≤ χ(H ) for any 3-uniform hypergraph H (assuming χ(H ) ≥ 2). The proof uses the list coloring version of Brooks’ theorem.