Poset Ramsey Numbers for Boolean Lattices

For each positive integer n, let Qn denote the Boolean lattice of dimension n. For posets P,P′\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$P, P^{\prime }$\end{document}, define the poset Ramsey numberR(P,P′)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$R(P,P^{\prime })$\end{document} to be the least N such that for any red/blue coloring of the elements of QN, there exists either a subposet isomorphic to P with all elements red, or a subposet isomorphic to P′\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$P^{\prime }$\end{document} with all elements blue. Axenovich and Walzer introduced this concept in Order (2017), where they proved R(Q2,Qn) ≤ 2n + 2 and R(Qn,Qm) ≤ mn + n + m. They later proved 2n ≤ R(Qn,Qn) ≤ n2 + 2n. Walzer later proved R(Qn,Qn) ≤ n2 + 1. We provide some improved bounds for R(Qn,Qm) for various n,m∈ℕ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$n,m \in \mathbb {N}$\end{document}. In particular, we prove that R(Qn,Qn) ≤ n2 − n + 2, R(Q2,Qn)≤53n+2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$R(Q_{2}, Q_{n}) \le \frac {5}{3}n + 2$\end{document}, and R(Q3,Qn)≤⌈3716n+5516⌉\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$R(Q_{3}, Q_{n}) \le \lceil \frac {37}{16}n + \frac {55}{16}\rceil $\end{document}. We also prove that R(Q2,Q3) = 5, and R(Qm,Qn)≤m−1+2m+1n+13m+2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$R(Q_{m}, Q_{n}) \le \left \lceil \left (m - 1 + \frac {2}{m+1} \right )n + \frac {1}{3} m + 2\right \rceil $\end{document} for all n > m ≥ 4.