On book-wheel Ramsey number (Q1586774)

Let \(K_n\) denote the complete graph on \(n\) vertices, \(C_n\) denote the circuit on \(n\) vertices, \(B_m\) denote the book \(K_2\vee K^c_m\) and \(W_n\) denote the wheel with \(n\) spokes. And let \(r(G,H)\) denote the smallest positive integer \(n\) so that for any 2-coloring of the edges of \(K_n\) either there is a copy of the graph \(G\) with all of its edges assigned the first color or a copy of the graph \(H\) with all of its edges assigned the second color. The author proves: \hskip 10mm (1) \(r(B_m,W_n)= 2n+1\) for \(m\geq 1\), \(n\geq 5m+ 3\); \hskip 10mm (2) \(r(B_m, K_2\vee C_n)= 2n+ 3\) for \(n\geq 9\) if \(m= 1\) or \(n\geq (m-1)(16m^3+ 16m^2- 24m- 10)+ 1\) if \(m\geq 2\).