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- Title
Off-diagonal book Ramsey numbers.
- Authors
Conlon, David; Fox, Jacob; Wigderson, Yuval
- 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 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 $c$.
- Subjects
RAMSEY numbers; RAMSEY theory
- Publication
Combinatorics, Probability & Computing, 2023, Vol 32, Issue 3, p516
- ISSN
0963-5483
- Publication type
Article
- DOI
10.1017/S0963548322000360