Generalized Ramsey numbers of cycles, paths, and hypergraphs

Given a k -uniform hypergraph G and a set of k -uniform hypergraphs H , the generalized Ramsey number f ( G , H , q ) is the minimum number of colors needed to edge-color G so that every copy of every hypergraph H ∈ H in G receives at least q different colors. In this note we obtain bounds, some asymptotically sharp, on several generalized Ramsey numbers, when G = K n or G = K n , n and H is a set of cycles or paths, and when G = K n k and H contains a clique on k + 2 vertices or a tight cycle.