Ramsey numbers of large books versus multipartite graphs

For graphs \(H_1\) $H_1$ and \(H_2\) $H_2$ , the Ramsey number \(r(H_1,H_2)\) $r(H_1,H_2)$ is the smallest positive integer N such that any graph G on N vertices contains \(H_1\) $H_1$ as a subgraph, or its complement contains \(H_2\) $H_2$ as a subgraph. Let \(B_{n}^{(k)}\) $B_n^{(k)}$ denote the book graph on \(n+k\) $n+k$ vertices which consists of n copies of \(K_{k+1}\) $K_{k+1}$ all sharing a common \(K_k\) $K_k$ , and let \(H:=K_p(a_1,\dots ,a_{p})\) $H:=K_p(a_1,\cdots ,a_p)$ be the complete p-partite graph with parts of sizes \(a_1,\dots ,a_{p}\) $a_1,\cdots ,a_p$ . Recently, strengthening a result of Fox, He and Wigderson (Adv. Combin. 4 (2023), 21pp), Fan and Lin (J. Combin. Theory Ser. A 199 (2023), 19pp) showed that for every \(k, p, t\ge 2\) $k,p,t\ge 2$ , there exists \(\delta >0\) $\delta >0$ such that the following holds for all large n. Let \(1\le a_1\le \dots \le a_{p-1}\le t\) $1\le a_1\le \cdots \le a_{p-1}\le t$ and \(a_{p}\le \delta n\) $a_p\le \delta n$ be positive integers. If \(a_1=1\) $a_1=1$ , then \(r(H, B^{(k)}_n)\le (p-1)(n+ka_2-1)+1\) $r(H,B_n^{(k)})\le (p-1)(n+k{a}_2-1)+1$ . The inequality is tight if \(n\equiv 1\pmod {a_2}\) $n\equiv 1(mod{a}_2)$ . In this paper, we improve the above upper bounds for the cases when \(n\equiv 2\pmod {a_2}\) $n\equiv 2(mod{a}_2)$ and \(n\equiv 3\pmod {a_2}\) $n\equiv 3(mod{a}_2)$ . Combining the new upper bounds and constructions of the lower bounds for these cases, we are able to determine the exact values of \(r(K_p(a_1,\dots ,a_{p}), B^{(k)}_n)\) $r(K_p(a_1,\cdots ,a_p),B_n^{(k)})$ when \(p=3\) $p=3$ . The bound on \(1/\delta \) $1/\delta$ we obtain is not of tower-type since our proof does not rely on the regularity lemma.