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.
Categories: math.CO