New upper bounds for Ramsey numbers (Q1266397)

New upper bound formulas for Ramsey numbers \(R(G_1,G_2)\) are obtained in terms of Ramsey numbers for subgraphs of \(G_1\) and \(G_2\). Using these formulas and known small order classical Ramsey numbers and generalized Ramsey numbers, new upper bounds are given for \(24\) new pairs of graphs, where each of these graphs is of order at most \(7\). For example, \(R(K_5,K_6) \leq 87\), and \(R(K_5-e,K_6-e) \leq 39\).