Ramsey numbers of Boolean lattices

Abstract

The poset Ramsey number (Formula presented.) is the smallest integer (Formula presented.) such that any blue–red coloring of the elements of the Boolean lattice (Formula presented.) has a blue-induced copy of (Formula presented.) or a red-induced copy of (Formula presented.). The weak poset Ramsey number (Formula presented.) is defined analogously, with weak copies instead of induced copies. It is easy to see that (Formula presented.). Axenovich and Walzer (Order 34 (2017), 287–298) showed that (Formula presented.). Recently, Lu and Thompson (Order 39 (2022), no. 2, 171–185) improved the upper bound to (Formula presented.). In this paper, we solve this problem asymptotically by showing that (Formula presented.). In the diagonal case, Cox and Stolee (Order 35 (2018), no. 3, 557–579) proved (Formula presented.) using a probabilistic construction. In the induced case, Bohman and Peng (arXiv preprint arXiv:2102.00317, 2021) showed (Formula presented.) using an explicit construction. Improving these results, we show that (Formula presented.) for all (Formula presented.) and large (Formula presented.) by giving an explicit construction; in particular, we prove that (Formula presented.). © 2023 The Authors. The publishing rights in this article are licensed to the London Mathematical Society under an exclusive licence.