Ramsey numbers of Boolean lattices

Grósz, Dániel and Methuku, Abhishek and Tompkins, Casey (2023) Ramsey numbers of Boolean lattices. BULLETIN OF THE LONDON MATHEMATICAL SOCIETY, First. ISSN 0024-6093


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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.

Item Type: Article
Additional Information: Export Date: 17 February 2023 Correspondence Address: Grósz, D.; Department of Mathematics, Italy; email: Funding details: Engineering and Physical Sciences Research Council, EPSRC, EP/S00100X/1 Funding details: Nemzeti Kutatási Fejlesztési és Innovációs Hivatal, NKFIH, K135800 Funding text 1: We thank the anonymous reviewer for the helpful comments and careful reading of our paper. The second and third authors were supported by the grant IBS‐R029‐C1. The research of the second author was also supported by the EPSRC, grant no. EP/S00100X/1 (A. Methuku), and the research of the third author was also supported by NKFIH grant K135800.
Subjects: Q Science / természettudomány > QA Mathematics / matematika
Depositing User: MTMT SWORD
Date Deposited: 24 Mar 2023 13:29
Last Modified: 24 Mar 2023 13:29

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