Off-diagonal book Ramsey numbers

The book graph B-a((k)) consists of n copies of Kk+1 joined along a common K-k. In the prequel to this paper, we studied the diagonal Ramsey number r(B-(k), B-(k)). Here we consider the natural off-diagonal variant r(B-n((k)), B-n((k))) for fixed c epsilon (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 nearlyextremal 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.