The connected size Ramsey number for matchings versus small disconnected graphs

Let F, G, and H be simple graphs. The notation F → (G, H) means that if all the edges of F are arbitrarily colored by red or blue, then there always exists either a red subgraph G or a blue subgraph H. The size Ramsey number of graph G and H, denoted by r̂(G, H) is the smallest integer k such that there is a graph F with k edges satisfying F → (G, H). In this research, we will study a modified size Ramsey number, namely the connected size Ramsey number. In this case, we only consider connected graphs F satisfying the above properties. This connected size Ramsey number of G and H is denoted by r̂c(G, H). We will derive an upper bound of r̂c(nK2, H), n ≥ 2 where H is 2Pm or 2K1, t, and find the exact values of r̂c(nK2, H), for some fixed n.