Exact Values and Bounds for Ramsey Numbers of $C_4$ Versus a Star Graph

Abstract: The 8 unknown values of the Ramsey numbers $R(C_4,K_{1,n})$ for $n \leq 37$ are determined, showing that $R(C_4,K_{1,27}) = 33$ and $R(C_4,K_{1,n}) = n + 7$ for $28 \leq n \leq 33$ or $n = 37$. Additionally, the following results are proven: $\bullet$ If $n$ is even and $\lceil\sqrt{n}\rceil$ is odd, then $R(C_4,K_{1,n}) \leq n + \left\lceil\sqrt{n-\lceil\sqrt{n}\rceil+2}\right\rceil + 1$. $\bullet$ If $m \equiv 2 \,(\text{mod } 6)$ with $m \geq 8$, then $R(C_4,K_{1,m^2+3}) \leq m^2 + m + 4$. $\bullet$ If $R(C_4,K_{1,n}) > R(C_4,K_{1,n-1})$, then $R(C_4,K_{1,2n+1-R(C_4,K_{1,n})}) \geq n$.