Polarity graphs and Ramsey numbers for [formula omitted] versus stars.

For two given graphs G 1 and G 2 , the Ramsey number R ( G 1 , G 2 ) is the smallest integer N such that for any graph of order N , either G contains a copy of G 1 or its complement contains a copy of G 2 . Let C m be a cycle of length m and K 1 , n a star of order n + 1 . Parsons (1975) shows that R ( C 4 , K 1 , n ) ≤ n + ⌊ n − 1 ⌋ + 2 and if n is the square of a prime power, then the equality holds. In this paper, by discussing the properties of polarity graphs whose vertices are points in the projective planes over Galois fields, we prove that R ( C 4 , K 1 , q 2 − t ) = q 2 + q − ( t − 1 ) if q is an odd prime power, 1 ≤ t ≤ 2 ⌈ q 4 ⌉ and t ≠ 2 ⌈ q 4 ⌉ − 1 , which extends a result on R ( C 4 , K 1 , q 2 − t ) obtained by Parsons (1976). [ABSTRACT FROM AUTHOR]