Ramsey numbers of Boolean lattices

Dániel Grósz, Abhishek Methuku, Casey Tompkins

Research output: Contribution to journalArticlepeer-review

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.).

Original languageEnglish (US)
Pages (from-to)914-932
Number of pages19
JournalBulletin of the London Mathematical Society
Volume55
Issue number2
DOIs
StatePublished - Apr 2023
Externally publishedYes

ASJC Scopus subject areas

  • General Mathematics

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