Rainbow Ramsey problems for the Boolean lattice

TL;DR

This paper investigates the minimal size of Boolean lattices needed to guarantee either a monochromatic or rainbow subposet under various coloring constraints, extending rainbow Ramsey theory to the Boolean lattice.

Contribution

It introduces new bounds and results for rainbow Ramsey problems in the Boolean lattice, considering both weak and strong versions, and related coloring restrictions.

Findings

Established bounds for rainbow Ramsey numbers in Boolean lattices

Analyzed weak and strong (induced and non-induced) versions of the problem

Explored related coloring problems avoiding rainbow antichains

Abstract

We address the following rainbow Ramsey problem: For posets $P,Q$ what is the smallest number $n$ such that any coloring of the elements of the Boolean lattice $B_n$ either admits a monochromatic copy of $P$ or a rainbow copy of $Q$. We consider both weak and strong (non-induced and induced) versions of this problem. We also investigate related problems on (partial) $k$-colorings of $B_n$ that do not admit rainbow antichains of size $k$.