Ramsey Numbers of $$C_4$$ versus Wheels and Stars.

Let $$ex(n, C_4)$$ denote the maximum size of a $$C_4$$ -free graph of order $$n$$ . For an even integer or odd prime power $$q$$ , we prove that $$ex(q^2+q+2,C_4)<\frac{1}{2}(q+1)(q^2+q+2)$$ , which leads to an improvement of the upper bound on Ramsey numbers $$R(C_4,W_{q^2+2})$$ , where $$W_n$$ is a wheel of order $$n$$ . By using a simple polarity graph $$G_q$$ for a prime power $$q$$ , we construct the graphs whose complements do not contain $$K_{1,m}$$ or $$W_m$$ , and then determine some exact values of $$R(C_4,K_{1,m})$$ and $$R(C_4,W_{m})$$ . In particular, we prove that $$R(C_4,K_{1, q^2-2})=q^2+q-1$$ for $$q\ge 3$$ , $$R(C_4,W_{q^2-1})=q^2+q-1$$ for $$q\ge 5$$ , and $$R(C_4,W_{q^2+2})=q^2+q+2$$ for $$q\ge 7$$ . [ABSTRACT FROM AUTHOR]