Strong Ramsey Numbers of Graphs

The Ramsey number R(F, H) of two graphs F and H without isolated vertices is the minimum positive integer n such that every red-blue coloring of the complete graph K n of order n results in a subgraph isomorphic to F all of whose edges are colored red or a subgraph isomorphic to H all of whose edges are colored blue.The (diagonal) Ramsey number R(F ) = R(F, F ) of F is the minimum positive integer n such that every red-blue coloring of K n results in a subgraph of K n isomorphic to F all of whose edges are colored the same (a monochromatic F ).The strong Ramsey number R(F, H) of two graphs F and H without isolated vertices is the minimum positive integer n such that every red-blue coloring of K n results in two edge-disjoint monochromatic subgraphs, one isomorphic to F and the other isomorphic to H. Strong Ramsey numbers R(F, H) have been determined for several pairs F, H of well-known graphs of small size.For many pairs F, H of graphs, it is shown that R(F, H) = max{R(F ), R(H)} while an example of two graphs F and H is given for which R(F, H) = 1 + max{R(F ), R(H)}, resulting in an open question.