Poset Ramsey Numbers for Boolean Lattices

TL;DR

This paper investigates the poset Ramsey numbers for Boolean lattices, providing improved bounds and exact values for specific cases, advancing understanding of colorings and subposet structures within Boolean lattices.

Contribution

The authors derive new upper bounds and exact values for poset Ramsey numbers involving Boolean lattices, improving upon previous results and covering a broader range of parameters.

Findings

Proved $R(Q_n, Q_n) ext{ bounds}$: $oxed{R(Q_n, Q_n) ext{ between } 2n ext{ and } n^2 + 2n}$.

Established $R(Q_2, Q_n) ext{ upper bound}$: $oxed{rac{5}{3}n + 2}$.

Determined $R(Q_2, Q_3) = 5$.

Abstract

A subposet $Q'$ of a poset $Q$ is a \textit{copy of a poset} $P$ if there is a bijection $f$ between elements of $P$ and $Q'$ such that $x \le y$ in $P$ iff $f(x) \le f(y)$ in $Q'$. For posets $P, P'$, let the \textit{poset Ramsey number} $R(P,P')$ be the smallest $N$ such that no matter how the elements of the Boolean lattice $Q_N$ are colored red and blue, there is a copy of $P$ with all red elements or a copy of $P'$ with all blue elements. Axenovich and Walzer introduced this concept in \textit{Order} (2017), where they proved $R(Q_2, Q_n) \le 2n + 2$ and $R(Q_n, Q_m) \le mn + n + m$, where $Q_n$ is the Boolean lattice of dimension $n$. They later proved $2n \le R(Q_n, Q_n) \le n^2 + 2n$. Walzer later proved $R(Q_n, Q_n) \le n^2 + 1$. We provide some improved bounds for $R(Q_n, Q_m)$ for various $n,m \in \mathbb{N}$. In particular, we prove that $R(Q_n, Q_n) \le n^2 - n + 2$, $R(Q_2, Q_n) \le \frac{5}{3}n + 2$, and $R(Q_3, Q_n) \le \frac{37}{16}n + \frac{39}{16}$. We also prove that $R(Q_2,Q_3) = 5$, and $R(Q_m, Q_n) \le (m - 2 + \frac{9m - 9}{(2m - 3)(m + 1)})n + m + 3$ for all $n \ge m \ge 4$.