Sub-Ramsey Numbers for Matchings

Given a graph G and a positive integer k, the sub-Ramsey number sr(G, k) is defined to be the minimum number m such that every K_m whose edges are colored using every color at most k times contains a subgraph isomorphic to G all of whose edges have distinct colors. In this paper, we will concentrate on sr(nK₂, k) with nK₂ denoting a matching of size n. We first give upper and lower bounds for sr(nK₂, k) and exact values of sr(nK₂, k) for some n and k. Afterwards, we show that sr(nK₂, k) = 2 n when n is sufficiently large and k<n8 by applying the Local Lemma.