Ramsey numbers of large books
Xun Chen, Qizhong Lin, Chunlin You

TL;DR
This paper proves an asymptotic upper bound for the Ramsey numbers of large books, confirming a longstanding conjecture and implications for related graph theory conjectures.
Contribution
It establishes an asymptotic bound for Ramsey numbers of large books, confirming Rousseau and Sheehan's conjecture.
Findings
Proves $r(B_m, B_n) o 2(m+n)$ as $n o
Confirms Rousseau and Sheehan's conjecture asymptotically
Implications for a conjecture on strongly regular graphs
Abstract
A book is a graph which consists of triangles sharing a common edge. In 1978, Rousseau and Sheehan conjectured that the Ramsey number satisfies for some constant . In this paper, we obtain that for all and large, which confirms the conjecture of Rousseau and Sheehan asymptotically. As a corollary, our result implies that a related conjecture of Faudree, Rousseau and Sheehan (1982) on strongly regular graph holds asymptotically.
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Taxonomy
TopicsLimits and Structures in Graph Theory · Advanced Topology and Set Theory · Advanced Graph Theory Research
