Large book--cycle Ramsey numbers

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