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 s r (n K 2 , k) with n K 2 denoting a matching of size n. We first give upper and lower bounds for s r (n K 2 , k) and exact values of s r (n K 2 , k) for some n and k. Afterwards, we show that s r (n K 2 , k) = 2 n when n is sufficiently large and k < n 8 by applying the Local Lemma. [ABSTRACT FROM AUTHOR]