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 determine the exact value of $r(B_n^{(2)}, C_m)$ for $\frac{9n}{10}\le m\le \frac{10n}{9}$ and $n$ large enough. This gives an answer to a question by Faudree, Rousseau and Sheehan in a stronger form when $m$ and $n$ are large. Furthermore, we are able to determine the asymptotic value of $r(B_n^{(k)}, C_n)$ for each fixed integer $k \geq 3$. Namely, we prove for each $k \geq 3$, $$r(B_n^{(k)}, C_n)= (k+1+o(1))n.$$ The proofs are mainly built upon results on the (weakly) pancyclic properties of graphs and a refined version of regularity lemma by Conlon.

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