Large book--cycle Ramsey numbers

TL;DR

This paper determines exact and asymptotic values of cycle-book Ramsey numbers for large parameters, extending previous results and answering longstanding questions in combinatorics.

Contribution

It provides the exact value of $r(B_n^{(2)}, C_m)$ for certain ranges and the asymptotic value of $r(B_n^{(k)}, C_n)$ for all $k \\geq 3$, advancing understanding of cycle-book Ramsey numbers.

Findings

Exact value of $r(B_n^{(2)}, C_m)$ for specified ranges.

Asymptotic value of $r(B_n^{(k)}, C_n)$ for all $k \\geq 3$.

Extension of previous results and resolution of a question from 1991.

Abstract

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 $n \geq 1000$. 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 $k \geq 3$. Namely, we prove that for each $k \geq 3$, $r(B_n^{(k)}, C_n)= (k+1+o_k(1))n.$ 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).