Rainbow ramsey numbers

We investigate a new generalization of the generalized ramsey number for graphs. Recall that the generalized ramsey number for graphs G₁, G₂,…, G_c is the minimum positive integer N such that any coloring of the edges of the complete graph K_N with c colors must contain a subgraph isomorphic to G_i in color i for some i. Bialostocki and Voxman defined RM(G) for a graph G to be the minimum N such that any edge-coloring of K_N with any number of colors must contain a subgraph isomorphic to G in which either every edge is the same color (a monochromatic G) or every edge is a different color (a rainbow G). This number exists if and only if G is acyclic.; Expanding on this definition, we define the rainbow ramsey number RR(G₁, G ₂) of graphs G₁ and G ₂ to be the minimum N such that any edge-coloring of K_N with any number of colors contains either a monochromatic G₁ or a rainbow G₂. This number exists if and only if G₁ is a star or G₂ is acyclic. We present upper and lower bounds for RR(K_1,n, K_m), RR(K_n, T_m), RR(K_n, K _m), RR(K_{1, n}, mK₂) and RR(nK ₂, K_1,m, where T_m is an arbitrary tree of order m.; We also define the edge-chromatic ramsey number CR(G₁,G₂) to be the minimum N such that any edge-coloring of K_N must contain either a monochromatic G ₁ or a properly edge-colored G₂. When both are defined, CR(G₁, G₂) ≤ RR(G₁ , G₂). We consider bounds for CR (C_n, P_m), CR( K_1,n, P_m), CR(P_n, P_m), and the corresponding rainbow ramsey numbers.; These two new ramsey numbers can be further generalized as the $F$-free ramsey number. For a set of graphs $F$, an $F$-free coloring of a graph G is a coloring so that G does not contain any monochromatic subgraph isomorphic to any graph in $F$. The $F$-free ramsey number of graphs G ₁ and G₂, denoted $RF$ (G₁, G₂), is the minimum N such that every edge-coloring of K_N contains either a monochromatic copy of G