Large book--cycle Ramsey numbers
Qizhong Lin, Xing Peng

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 be the book graph which consists of copies of all sharing a common , and let be a cycle of length . In this paper, we first determine the exact value of for and . 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 and are large. Building upon this exact result, we are able to determine the asymptotic value of for each . Namely, we prove that for each , 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).
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Taxonomy
TopicsLimits and Structures in Graph Theory · Advanced Topology and Set Theory · Advanced Graph Theory Research
