Off-diagonal book Ramsey numbers
David Conlon, Jacob Fox, Yuval Wigderson

TL;DR
This paper explores off-diagonal Ramsey numbers for book graphs, revealing a dichotomy where small c values favor structured colorings and larger c values favor quasirandom colorings, with open questions for intermediate c.
Contribution
It introduces the off-diagonal Ramsey number for book graphs and characterizes the extremal colorings, showing a transition from structured to quasirandom behavior as c varies.
Findings
For small c, k-partite constructions are nearly extremal.
For larger c, random colorings are asymptotically optimal.
Nearly-extremal colorings are either close to k-partite or quasirandom.
Abstract
The book graph consists of copies of joined along a common . In the prequel to this paper, we studied the diagonal Ramsey number . Here we consider the natural off-diagonal variant for fixed . In this more general setting, we show that an interesting dichotomy emerges: for very small , a simple -partite construction dictates the Ramsey function and all nearly-extremal colorings are close to being -partite, while, for bounded away from , random colorings of an appropriate density are asymptotically optimal and all nearly-extremal colorings are quasirandom. Our investigations also open up a range of questions about what happens for intermediate values of .
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Taxonomy
TopicsLimits and Structures in Graph Theory · Analytic Number Theory Research · Advanced Topology and Set Theory
