Constrained Ramsey Numbers of Matchings

The rainbow Ramsey number \( RR(G_1, G_2) \) or constrained Ramsey number \( f(G_1,G_2) \) of two graphs \( G_1 \) and \( G_2 \) is defined to be the minimum integer \( N \) such that any edge-coloring of the complete graph \( K_N \) with any number of colors must contain either a subgraph isomorphic to \( G_1 \) with every edge the same color or a subgraph isomorphic to \( G_2 \) with every edge a different color. This number exists if and only if \( G_1 \) is a star or \( G_2 \) is acyclic. In this paper, we present the conjecture that the constrained Ramsey number of \( nK_2 \) and \( mK_2 \) is \( m(n-1)+2 \), along with a proof in the case \( m \leq \frac{3}{2}(n-1) \).