On some Ramsey numbers of C_4 versus K_(2,n)

摘要

For given graphs H_1, H_2, the Ramsey number R(H_1,H_2) is the smallest positive integer n such that if we arbitrarily color the edges of the complete graph K_n with two colors 1 (red) and 2 (blue), then there is monochromatic copy of H_1 colored with 1 or H_2 colored with 2. We show that if n is even, q =「n~(1/2)| is odd, and s = n- (q- 1)~2 ≤ q/2, then R(K_(2,2),K_(2,n)) ≤n + 2q-1, where K_(n,m) are complete bipartite graphs. The latter bound gives the exact value of R(K_(2,2),K_(2,18)) = 27. Moreover, we show that R(K_(2,2),K_(2,14)) = 22 and.R(K_(2,2).K_(2,15)) = 24.