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 \(sr(nK_{2},k)\) with \(nK_{2}\) denoting a matching of size n. We first give upper and lower bounds for \(sr(nK_{2},k)\) and exact values of \(sr(nK_{2},k)\) for some n and k. Afterwards, we show that \(sr(nK_{2},k)=2n\) when n is sufficiently large and \(k<\frac{n}{8}\) by applying the Local Lemma.

给定一个图G和一个正整数k，子拉姆西数sr（G，  k）被定义为最小数m，使得每个\（K_ {m} \）的边缘使用最多k种颜色进行着色包含一个与G同构的子图，所有图的边缘都有不同的颜色。在本文中，我们将集中讨论\（sr（nK_ {2}，k）\），其中\（nK_ {2} \）表示大小为n的匹配项。首先给出上部和下界\（SR（nK_ {2}，k）的\）和的精确值\（SR（nK_ {2}，k）的\）对于一些n和k。然后，通过应用局部引理，我们证明当n足够大时\（sr（nK_ {2}，k）= 2n \）和\（k <\ frac {n} {8} \）。