Ramsey numbers of long even cycles versus books
Qizhong Lin, Shixi Song
TL;DR
This paper resolves the exact values of g_k=R(C_m,B_n^{(k)}) for every fixed k≥3 under the regime of large even m and n with (t-1)(m-1) ≤ n-1 < t(m-1). It combines explicit lower-bound constructions with sophisticated semi-random, partition-based upper-bound analysis of Ramsey graphs, building on a suite of classical and tailor-made lemmas to control the structure of potential Ramsey graphs. The key contribution is a complete determination of g_k across three regimes (i)-(iii), extending prior results for k=1,2 to all k≥3 under the same asymptotic conditions. The approach blends combinatorial constructions, extremal graph theory, and careful decomposition arguments to yield tight upper bounds and sharp lower bounds, advancing our understanding of how long even cycles interact with books in Ramsey theory.
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 $m = Ω(n^{9/10})$. 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 $k\ge3$ 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.