On Ramsey numbers of fans (Q1003775)

The graph \(F_n = K_1 + nK_2\) is the Fan graph, and consists of \(n\) triangles sharing a single vertex. It was shown in [\textit{Y. Li} and \textit{C.C. Rousseau}, J. Graph Theory 23, No.\,4, 413--420 (1996; Zbl 0954.05031)] that the Ramsey number \(r(F_1, F_n) = 4n + 1\) for \(n \geq 2\), and this result is extended by showing that \(r(F_2, F_n) = 4n + 1\) also for \(n \geq 2\). In addition, the general upper bound \(r(F_s, F_n) \leq 4n + 2s\) for \(2 \leq s \leq n\) is proved.