Boolean lattices: Ramsey properties and embeddings

TL;DR

This paper investigates the Ramsey properties of Boolean lattices, establishing bounds on the poset Ramsey number and the count of sublattice copies, advancing understanding of colorings and embeddings in these structures.

Contribution

It provides new bounds on the poset Ramsey number for Boolean lattices and asymptotically tight estimates for the number of sublattice copies within larger Boolean lattices.

Findings

Established bounds on the poset Ramsey number R(P,P') for Boolean lattices.

Derived asymptotically tight bounds for the number of copies of Q_n in Q_N.

Extended results to a multicolor version of the poset Ramsey number.

Abstract

A subposet $Q'$ of a poset $Q$ is a copy of a poset $P$ if there is a bijection $f$ between elements of $P$ and $Q'$ such that $x\leq y$ in $P$ iff $f(x)\leq f(y)$ in $Q'$. For posets $P, P'$, let the 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. We provide some general bounds on $R(P,P')$ and focus on the situation when $P$ and $P'$ are both Boolean lattices. In addition, we give asymptotically tight bounds for the number of copies of $Q_n$ in $Q_N$ and for a multicolor version of a poset Ramsey number.