Strong Ramsey Numbers of Graphs of Size 4

The Ramsey number R(G) of a graph G without isolated vertices is the minimum order n of the complete graph Kn for which every red-blue edge coloring results in a monochromatic subgraph isomorphic to G. The strong Ramsey number R¯(F,H) of two non-isomorphic graphs F and H without isolated vertices is the minimum positive integer n such that every red-blue coloring of Kn results in two edge-disjoint monochromatic subgraphs, one isomorphic to F and the other isomorphic to H. The numbers R¯(F,H) have been determined for all pairs F,H of non-isomorphic graphs F and H of size 3. Here, R¯(F,H) is determined for all pairs F,H of non-isomorphic graphs F and H of size 4. In each case, it is shown that R¯(F,H) is either max{R(F),R(H)} or 1+max{R(F),R(H)}. Furthermore, R¯(F,H) has been determined for some special pairs F,H of non-isomorphic graphs.