Off-diagonal book Ramsey numbers

Understand

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.