Large Book-Cycle Ramsey Numbers

Let $B_n^{(k)}$ be the book graph which consists of $n$ copies $K_{k+1}$ all sharing a common $K_k$, and let $C_m$ cycle length $m$. In this paper, we first determine exact value $r(B_n^{(2)}, C_m)$ for $\frac{8}{9}n+112\le m\le \lceil\frac{3n}{2}\rceil+1$ $n \geq 1000$. This answers question Faudree, Rousseau, Sheehan [Ars Combin., 31 (1991), pp. 239--248] in stronger form when $m$ are large. Building upon result, able to asymptotic $r(B_n^{(k)}, C_n)$ each $k 3$. Namely, prove that 3$, C_n)= (k+1+o_k(1))n$. extends result due Rousseau [J. London Math. Soc., 18 (1978), 392--396].