Summary: For graphs \(F\), \(G\) and \(H\), we write \(F\rightarrow (G,H)\) to mean that if the edges of \(F\) are colored with two colors, say red and blue, then the red subgraph contains a copy of \(G\) or the blue subgraph contains a copy of \(H\). The graph \(F\) is called a Ramsey \((G,H)\) graph if \(F\rightarrow (G,H)\). Furthermore, the graph \(F\) is called a Ramsey \((G,H)\)-minimal graph if \(F\rightarrow(G,H)\) but \(F-e\not\rightarrow (G,H)\) for any edge \(e\in E(F)\). In this paper, we characterize all unicyclic Ramsey \((G,H)\)-minimal graphs when \(G\) is a matching \(mK_2\) for any integer \(m\geq 2\) and \(H\) is a path on four vertices.