Some values of Ramsey numbers for C4 versus stars.

For two given graphs G1 and G2, the Ramsey number R(G1,G2) is the smallest integer N such that for any graph of order N, either G contains a copy of G1 or its complement contains a copy of G2. Let Cm be a cycle of length m and K1,n a star of order n+1. Parsons (1975) [6] shows that R(C4,K1,n)n+n1+2 for all n2 and the equality holds if n is the square of a prime power. Let q be a prime power. In this paper, we first construct a graph q on q21 vertices without C4 by using the Galois field Fq, and then we prove that R(C4,K1,(q1)2+t)=(q1)2+q+t for q4 is even and t=1,0,2, and R(C4,K1,q(q1)t)=q2t for q5 is odd and t=2,4,...,2q4.