Computation of the Ramsey Numbers R(C4,K9)R(C_4,K_9) and R(C4,K10)R(C_4,K_{10})

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