Two types of size Ramsey numbers for matchings of small order

For simple graphs G and H, their size Ramsey number (Figure presented.) is the smallest possible size of F such that for any red-blue coloring of its edges, F contains either a red G or a blue H. Similarly, we can define the connected size Ramsey number (Figure presented.) by adding the prerequisite that F must be connected. In this paper, we explore the relationships between these size Ramsey numbers and give some results on their values for certain classes of graphs. We are mainly interested in the cases where G is either a 2K ₂ or a 3K ₂, and where H is either a cycle C_n or a union of paths nP_m. Additionally, we improve an upper bound regarding the values of (Figure presented.) and (Figure presented.) for certain t and m.