Boolean lattice without small rainbow subposets

TL;DR

This paper investigates the structure of colorings in Boolean lattices that avoid small rainbow subposets, providing exact values and bounds for related Ramsey numbers, thus advancing understanding in poset Ramsey theory.

Contribution

It introduces new structural insights into Boolean lattice colorings avoiding small rainbow subposets and determines exact and bounded values for Boolean Gallai-Ramsey and rainbow Ramsey numbers.

Findings

Exact values for Boolean Gallai-Ramsey numbers

Bounds for Boolean rainbow Ramsey numbers

Improved results over previous studies

Abstract

A Boolean lattice $\mathcal{B}_n=(2^X, \leq)$ is the power set of an $n$-element ground set $X$ equipped with inclusion relation. For two posets $\mathcal{P}$ and $\mathcal{Q}$, we say that $\mathcal{Q}$ contains an \emph{induced copy} of $\mathcal{P}$ if there exists an injection $f : \mathcal{P} \to \mathcal{Q}$ such that $f(X) \le f(Y)$ if and only if $X \le Y$ in $\mathcal{P}$. A $k$-coloring is exact if all colors are used at least once. For posets $\mathcal{Q}$ and $\mathcal{P}$, the \emph{Boolean Gallai-Ramsey number} $\operatorname{GR}_{k}(\mathcal{Q}:\mathcal{P})$ is defined as the smallest $n$ such that any exact $k$-coloring of the sets in $\mathcal{B}_n$ contains either a rainbow induced copy of $\mathcal{Q}$ or a monochromatic induced copy of $\mathcal{P}$ and the \emph{Boolean rainbow Ramsey number} $\operatorname{RR}(\mathcal{Q}:\mathcal{P})$ is defined as the smallest $n$ such that any coloring of the sets in $\mathcal{B}_n$ contains either a rainbow induced copy of $\mathcal{Q}$ or a monochromatic induced copy of $\mathcal{P}$. In this paper, we first study the structural properties of exact $k$-colorings of the sets in Boolean lattice without rainbow induced copy of small posets. As the application of these results, we give exact values and some bounds of Boolean Gallai-Ramsey numbers and Boolean rainbow Ramsey numbers, which improve a result of Chen, Cheng, Li, and Liu in 2020 and give an answer of a question proposed by Chang, Gerbner, Li, Methuku, Nagy, Patk\'{o}s, and Vizer in 2022.