New Upper and Lower Bounds for Ramsey Numbers

A graphG of order p is called a(G1,G2; p)-graph ((m,n; p)-graph, respectively) if G does not containG1 and G does not containG2 (Km and Kn, respectively). It is easy to see thatR(G1,G2) = p0 + 1 iff p0 =max {p| there exists a(G1,G2 : p)-graph}. In this paper, f (G1) (g(G2), respectively) denotes the number of G1 (G2, respectively) inG (G, respectively) as a subgraph. The R(G1,G2; p)-graph is called a(G1,G2; p)-Ramsey graph if p = R(G1,G2)−1. Letdi be the degree of vertex i in G of orderp, and letdi = p−1−di , where 1≤ i ≤ p. If G, H are graphs, G◦H denotes a{G∨H,G+H}-graph, where ‘∨’ is the join operation (see [1]). Let Gk i (i = 1,2) bea graph with order k and letG1 = G m−s 1 ◦ G s 1, G2 = G n−t 2 ◦G t 2. Taking any vertex (y, respectively), let G s+1 1 = {x}◦G s 1, G t+1 2 = {y}◦G t 2. The number ofG1 (G t 2, respectively) inG s+1 1 (G t+1 2 , respectively) as a subgraph is denoted by as (bt , respectively). Thus we have the following theorem. THEOREM 1 ([2]). For any(G1,G2; p)-graph,the following inequalities must hold: as f (G s+1 1 ) ≤ f (G s 1)[R(G m−s 1 ,G2)− 1] (1) bt g(G t+1 2 ) ≤ g(G t 2)[R(G1,G n−t 2 )− 1]. (2)