On Some Ramsey Numbers of \(C_{4}\) versus \(K_{2,n}\)

For given graphs \( H_1 \) and \( H_2 \), the \emph{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 a monochromatic copy of \( H_1 \) colored with 1 or \( H_2 \) colored with 2.

We show that if \( n \) is even, \( q = \lceil \sqrt{n} \rceil \) is odd, and \( s = n – (q-1)^2 \leq \frac{q}{2} \), then \( R(K_{2,2}, K_{2,n}) \leq n + 2q – 1 \), where \( K_{n,m} \) are complete bipartite graphs. This bound provides 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 \( R(K_{2,2}, K_{2,15}) = 24 \).