Ramsey numbers of long even cycles versus books

Abstract

For any positive integers $k$ and $n$, let $B_n^{(k)}$ be the book graph consisting of $n$ copies of the complete graph $K_{k+1}$ sharing a common $K_k$. Let $C_m$ be a cycle of length $m$. Prior work by Allen, Łuczak, Polcyn, and Zhang (2023) established the Ramsey number $R(C_m,B_n^{(1)})$ for all sufficiently large even integer $m=\Omega (n^{9/10})$. Recently, Hu, Lin, Łuczak, Ning, and Peng (2025) obtained the exact value of $R(C_m,B_n^{(2)})$ under the same asymptotic conditions. A natural problem is to determine the exact value of $R(C_m,B_n^{(k)})$ for each fixed $k\ge 3$ under similar conditions. This paper provides a complete solution to this problem. The lower bound is proved by an explicit construction, while the tight upper bound is established by analyzing the corresponding Ramsey graph using semi-random ideas.

18 pages