New Upper Bounds for Ramsey Numbers

The problem of determining Ramsey numbers is known to be very difficult. The few known exact values and several bounds for different G1, G2 or m, n are scattered among many technical papers (see [3]). A graph G with order p is called a (G1,G2; p)-graph ((m, n; p)-graph, resp.) if G does not contain a G1 and Ḡ does not contain a G2 (Km and Kn , resp.). It is easy to see that R(G1,G2) = p0 +1 iff p0 = max{p | there exists a (G1,G2; p)-graph}. In this paper, f (G1) (g(G2), resp.) denotes the number of G1 (G2, resp.) in G (Ḡ, resp.) as a subgraph. The (G1,G2; p)-graph is called a (G1,G2; p)-Ramsey graph if p = R(G1,G2)− 1. Let di be the degree of vertex i in G of order p, and let d̄i = p − 1 − di , where 1 ≤ i ≤ p. If G, H are graphs, G ◦H denotes one of {G∨H,G+H}-graph, where ‘∨’ is the join operation (see [1]). Let Gk i (i = 1, 2) be a graph with order k and let G1 = Gm−s 1 ◦G1, G2 = Gn−t 2 ◦G2. Taking any vertex x (y, resp.), let Gs+1 1 = {x} ◦G1, Gt+1 2 = {y} ◦G2. The number of G1 (G2, resp.) in Gs+1 1 (G t+1 2 , resp.) as a subgraph is denoted by as (bt , resp.). Thus we have: