Ramsey Numbers of C4 versus Wheels and Stars

Let ex(n,C4) denote the maximum size of a C4-free graph of order n. For an even integer or odd prime power q > 6, we prove that ex(q2 + q + 2, C4) < 1 2(q+1)(q 2+q+2), which leads to an improvement of the upper bounds on Ramsey numbers R(C4,Wq2+2), where Wn is a wheel of order n. By using a simple polarity graph Gq for a prime power q, we construct the graphs whose complements do not contain K1,m or Wm, and then determine some exact values of R(C4,K1,m) and R(C4,Wm). In particular, we prove that R(C4,K1,q2−2) = q 2 + q − 1 for q > 3, R(C4,Wq2−1) = q 2 + q − 1 for q > 5, and R(C4,Wq2+2) = q2 + q + 2 for q > 7.