An edge colored graph is called a rainbow if no two of its edges have the same color. Let H and g be two families of graphs. Denote by RM (H,g) the smallest integer R, if it exists, having the property that every coloring of the edges of K_R by an arbitrary number of colors implies that either there is a monochromatic subgraph of K_R that is isomorphic to a graph in H, or there is a rainbow subgraph of K_R that is isomorphic to a graph in g. If there is no integer R that satisfies the property above, then we write RM(H,g) = ∞. If one of the sets H or g contains only a single graph H or G, respectively, then we use the simplified notation RM(H,G) or RM(H,g) or RM(H,G). For a family of graphs H and an integer s, we denote by r(H,s) the corresponding Ramsey number, which is defined to be the smallest integer r such that every s-coloring of the edges of K_r implies the existence of a monochromatic subgraph of K_r that is isomorphic to a graph in H. If H contains only a single graph H, then we denote r(H,s) by r(H,s).
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