Computation of the Ramsey Numbers R(C_4, K_9) and R(C_4, K_(10))

摘要

The Ramsey number R(C_4, K_m) is the smallest n such that any graph on n vertices contains a cycle of length four or an independent set of order m. With the help of computer algorithms we obtain the exact values of the Ramsey numbers R(C_4,K_9) = 30 and R(C_4, K_(10)) = 36. New bounds for the next two open cases are also presented.