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arXiv · 2110.14483

Off-diagonal book Ramsey numbers

David Conlon·Jacob Fox·Yuval Wigderson
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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 $c$.

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David Conlon, Jacob Fox, Yuval Wigderson. 2021-10-27. Off-diagonal book Ramsey numbers. https://arxiv.org/abs/2110.14483

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