Boolean lattice without small rainbow subposets

Abstract

A Boolean lattice ${\mathcal{B}}_n=(2^X,\le )$ 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} ${\mathrm{GR}}_k(\mathcal{Q}:\mathcal{P})$ 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} $\mathrm{RR}(\mathcal{Q}:\mathcal{P})$ 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, Patkós, and Vizer in 2022.