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=Kₙ or G=Kn,n and H is a set of cycles or paths, and when G=Kₙᵏ and H contains a clique on $k+2$ vertices or a tight cycle.