For graphs \(G_1\) and \(G_2\) , the Ramsey number \(R(G_1, G_2)\) is the smallest integer n such that every graph of order n either contains \(G_1\) as a subgraph or its complement contains \(G_2\) as a subgraph. A wheel \(W_m\) is formed by connecting a vertex to every vertex of a cycle \(C_m\) , while a fan \(F_n\) consists of n triangles sharing a common vertex. Hao and You (2023) established that \(R(W_4, F_n) = 4n+1\) for sufficiently large n, where the required lower bound on n is of exponential tower type, a consequence of their reliance on the Erdős-Simonovits stability theorem. We significantly improve this result by proving that the same formula holds for \(n \ge 111\) . Moreover, by applying the stability theorem, we further generalize this result by replacing \(W_4\) with \(W_{2m}\) , proving that for \(m \ge 2\) and large n, \(R(W_{2m}, F_n)=4n+m-\mu \) , where \(\mu =1\) if m is even, and \(\mu =0\) if m is odd. As a final extension, we consider the class \(\mathcal {W}\) of all wheels and show that \(R(\mathcal {W}, F_n)=4n+1\) for all \(n\ge 16.\)
