Subgraph Counting Identities and Ramsey Numbers

For each vertexvof a graphG, we consider the numbers of subgraphs of each isomorphism class which lie in the neighbourhood or complementary neighbourhood ofv. These numbers, summed overv, satisfy a series of identities that generalise some previous results of Goodman and ourselves. As sample applications, we improve the previous upper bounds on two Ramsey numbers. Specifically, we show thatR(5, 5)⩽49 andR(4, 6)⩽41. We also give some experimental evidence in support of our conjecture thatR(5, 5)=43.