Size Ramsey numbers of small graphs versus fans or paths

For two graphs G₁ and G₂, the size Ramsey number r (G₁, G₂) is the smallest positive integer m for which there exists a graph G of size m such that for any red-blue edge-coloring of the graph G, G contains either a red subgraph isomorphic to G₁, or a blue subgraph isomorphic to G₂. Let Pₙ be a path with n vertices, nK₂ a matching with n edges, and Fₙ a graph with n triangles sharing exactly one vertex. If G₁ is a small fixed graph and G₂ denotes any graph from a graph class, one can sometimes completely determine r (G₁, G₂). Faudree and Sheehan confirmed all size Ramsey numbers of P₃ versus complete graphs in 1983. The next year Erdos and Faudree confirmed that of 2K₂ versus complete graphs and complete bipartite graphs. We obtain three more Ramsey results of this type. For n 3, we prove that r (P₃, Fₙ) =4n+4 if n is odd, and r (P₃, Fₙ) =4n+5 if n is even. This result refutes a conjecture proposed by Baskoro et al. We also show that r (2K₂, F₂) =12 and r (2K₂, Fₙ) =5n+3 for n 3. In addition, we prove that r (2K₂, nPₘ) =\nm+1, (n+1) (m-1) \. This result verifies a conjecture posed by Vito and Silaban.