Poset Ramsey Numbers for Boolean Lattices

Author

Joshua Cain Thompson

Date of Award

Summer 2022

Document Type

Open Access Dissertation

Department

Mathematics

First Advisor

Linyuan Lu

Abstract

For each positive integer n, let Qn denote the Boolean lattice of dimension n. For posets P, P', define the poset Ramsey number R(P,P') to be the least N such that for any red/blue coloring of the elements of Q_N, there exists either a subposet isomorphic to P with all elements red, or a subposet isomorphic to P' with all elements blue.

Axenovich and Walzer introduced this concept in Order (2017), where they proved R(Q₂, Q_n) ≤ 2n + 2 and R(Q_n, Q_m) ≤ mn + n + m. They later proved 2n ≤R(Q_n, Q_n) ≤ n₂ + 2n. Walzer later proved R(Q_n, Q_n) ≤ n₂ + 1. We provide some improved bounds for R(Q_n, Q_m) for various n,m ∈ u+2115. In particular, we prove that R(Qn, Qn) ≤ n^₂ - n + 2, R(Q₂, Q_n) ≤ 5/3n + 2, and R(Q₃ , Q_n) ≤ ⌈ 37/16n + 55/16 ⌉. We also prove that R(Q₂,Q₃) = 5, and R(Q_m, Q_n) ≤ ⌈(m - 1 + 2/(m+1))n + 1/3 m + 2 ⌉ for all n > m ≥ 4.

Rights

© 2022, Joshua Cain Thompson

Recommended Citation

Thompson, J. C.(2022). Poset Ramsey Numbers for Boolean Lattices. (Doctoral dissertation). Retrieved from https://scholarcommons.sc.edu/etd/6931