A Construction for Boolean cube Ramsey numbers

TL;DR

This paper presents an explicit construction proving that the Boolean cube Ramsey number R(Q_n,Q_n) is at least 2n+1 for all n≥3, improving the understanding of these combinatorial bounds.

Contribution

The authors provide the first explicit construction establishing the lower bound R(Q_n,Q_n)≥2n+1 for all n≥3, advancing previous probabilistic results.

Findings

Established R(Q_n,Q_n)≥2n+1 for all n≥3.

Improved the lower bound from probabilistic methods to an explicit construction.

Bounded the Ramsey number between 2n+1 and n^2 - 2n + 2.

Abstract

Let $Q_n$ be the poset that consists of all subsets of a fixed $n$-element set, ordered by set inclusion. The poset cube Ramsey number $R(Q_n,Q_n)$ is defined as the least $m$ such that any 2-coloring of the elements of $Q_m$ admits a monochromatic copy of $Q_n$. The trivial lower bound $R(Q_n,Q_n)\ge 2n$ was improved by Cox and Stolee, who showed $R(Q_n,Q_n)\ge 2n+1$ for $3\le n\le 8$ and $n\ge 13$ using a probabilistic existence proof. In this paper, we provide an explicit construction that establishes $R(Q_n,Q_n)\ge 2n+1$ for all $n\ge 3$. The best known upper bound, due to Lu and Thompson, is $ R(Q_n, Q_n) \le n^2 - 2n + 2$.