Generalized Ramsey numbers via conflict-free hypergraph matchings

Given graphs G, H and an integer q 2, the generalized Ramsey number, denoted r (G, H, q), is the minimum number of colours needed to edge-colour G such that every copy of H receives at least q colours. In this paper, we prove that for a fixed integer k 3, we have r (Kₙ, Cₖ, 3) = n/ (k-2) +o (n). This generalises work of Joos and Muybayi, who proved r (Kₙ, C₄, 3) = n/2+o (n). We also provide an upper bound on r (K₍, ₍, Cₖ, 3), which generalises a result of Joos and Mubayi that r (K₍, ₍, C₄, 3) = 2n/3+o (n). Both of our results are in fact specific cases of more general theorems concerning families of cycles.