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 maxR (F), R (H) or 1+maxR (F), R (H). Furthermore, R¯ (F, H) has been determined for some special pairs F, H of non-isomorphic graphs.