Off-diagonal book Ramsey numbers

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 $B_n^{(k)}$ consists of $n$ copies of $K_{k+1}$ joined along a common $K_k$. In the prequel to this paper, we studied the diagonal Ramsey number $r(B_n^{(k)}, B_n^{(k)})$. Here we consider the natural off-diagonal variant $r(B_{cn}^{(k)}, B_n^{(k)})$ for fixed $c \in (0,1]$. In this more general setting, we show that an interesting dichotomy emerges: for very small $c$, a simple $k$-partite construction dictates the Ramsey function and all nearly-extremal colorings are close to being $k$-partite, while, for $c$ bounded away from $0$, 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 $c$.