Boolean lattice without small rainbow subposets

TL;DR

The paper tackles the Boolean-lattice Ramsey landscape by deriving structural characterizations of exact \\$k\\-colorings of \\$\\mathcal{B}_n\\$ that avoid rainbow induced copies of small posets such as \\$\\mathcal{C}_3\\$, \\$\\vee_2\\$, and \\$\\mathcal{B}_2\\$. These structural theorems enable precise determinations and bounds for Boolean Gallai-Ramsey numbers and Boolean rainbow Ramsey numbers, yielding exact values like \\$GR_k(\\mathcal{C}_3:\\mathcal{C}_s) = s\\$ when 3 ≤ \\$k\\≤ \\$ {s-1 \choose ceil((s-1)/2)} + 1\\$, and establishing goodness thresholds for larger \\$k\\$. For the fork and 2D Boolean lattice, the authors obtain \\$GR_3(\\vee_2:\\mathcal{C}_s) = 2s-1\\$ and related results, and give tight-type classifications for colorings to bound \\$GR_k(\\mathcal{B}_2:\\mathcal{B}_n)\\$ in terms of the classical Ramsey numbers \\$R_3(\\mathcal{B}_n)\\$. In the rainbow Ramsey setting, they prove a sharp upper bound \\$\\operatorname{RR}(\\mathcal{B}_m:\\mathcal{B}_n) \le m \, \\operatorname{R}_{2^m-1}(\\mathcal{B}_n) + m\\$, improving previous dependence on \\$m\\$ and unifying methods via the colored-structure framework; for the case \\$m=2\\$, they connect to \\$R_3(\\mathcal{B}_n)\\. Overall, the work advances Boolean Ramsey theory by translating rainbow-forbidden patterns into precise color-partition structures, solving open questions and strengthening links between Gallai-Ramsey and rainbow Ramsey paradigms in posets.

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, Patkós, and Vizer in 2022.

Theorems & Definitions (53)

• Definition 1
• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Theorem 1.6
• Theorem 2.1
• proof
• Corollary 2.2