Off-diagonal book Ramsey numbers

Abstract

AbstractThe 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 colourings are close to being$k$-partite, while, for$c$bounded away from$0$, random colourings of an appropriate density are asymptotically optimal and all nearly-extremal colourings are quasirandom. Our investigations also open up a range of questions about what happens for intermediate values of$c$.