Edge-colored 3-uniform hypergraphs without rainbow paths of length 3 and its applications to Ramsey theory

TL;DR

This work characterizes edge-colored complete 3-uniform hypergraphs that avoid rainbow copies of the three length-3 paths $\mathcal{T}$ (tight), $\mathcal{M}$ (messy), and $\mathcal{L}$ (loose). The authors prove sharp structural theorems: rainbow-$\mathcal{T}$-free colorings admit a partition-based description with color-dominant edges, rainbow-$\mathcal{M}$-free colorings have bounded color diversity, and rainbow-$\mathcal{L}$-free colorings exhibit strong removal properties yielding near-monochromatic subgraphs; these culminate in Ramsey-type results that relate constrained Ramsey numbers to the two-color Ramsey numbers, and precise anti-Ramsey numbers for each path. They extend these results to a multipartite setting via canonical Ramsey theory, obtaining analogous structural decompositions and exact multipartite anti-Ramsey values, and they provide a suite of existence results for multipartite constrained Ramsey numbers. The paper thus connects rainbow path avoidance in 3-uniform hypergraphs to classical Ramsey theory, delivering both structural characterizations and sharp quantitative outcomes with broad applicability to hypergraph Ramsey theory and its multipartite generalizations.

Abstract

Motivated by Ramsey theory problems, we consider edge-colorings of 3-uniform hypergraphs that contain no rainbow paths of length 3. There are three 3-uniform paths of length 3: the tight path $\mathcal{T}=\{v_1v_2v_3, v_2v_3v_4, v_3v_4v_5\}$, the messy path $\mathcal{M}=\{v_1v_2v_3, v_2v_3v_4, v_4v_5v_6\}$ and the loose path $\mathcal{L}=\{v_1v_2v_3,$ $v_3v_4v_5, v_5v_6v_7\}$. In this paper, we characterize the structures of edge-colored complete 3-uniform hypergraph $K_n^{(3)}$ without rainbow $\mathcal{T}$, $\mathcal{M}$ and $\mathcal{L}$, respectively. This generalizes a result of Thomason-Wagner on edge-colored complete graph $K_n$ without rainbow paths of length 3. We also obtain a multipartite generalization of these results. As applications, we obtain several Ramsey-type results. Given two $3$-uniform hypergraphs $H$ and $G$, the {\it constrained Ramsey number} $f(H,G)$ is defined as the minimum integer $n$ such that, in every edge-coloring of $K^{(3)}_n$ with any number of colors, there is either a monochromatic copy of $H$ or a rainbow copy of $G$. For $G\in \{\mathcal{T}, \mathcal{M}, \mathcal{L}\}$ and infinitely many 3-uniform hypergraphs $H$, we reduce $f(H, G)$ to the 2-colored Ramsey number $R_2(H)$ of $H$, that is, $f(H, G)=R_2(H)$. Given a $3$-uniform hypergraph $G$ and an integer $n\geq |V(G)|$, the {\it anti-Ramsey number} $ar(n, G)$ is the minimum integer $k$ such that, in every edge-coloring of $K^{(3)}_n$ with at least $k$ colors, there is a rainbow copy of $G$. We show that $ar(n, \mathcal{T})=\left\lfloor\frac{n}{3}\right\rfloor+2$ for $n\geq 5$, $ar(n, \mathcal{M})=3$ for $n\geq 7$, and $ar(n, \mathcal{L})=n$ for $n\geq 7$. Our newly obtained Ramsey-type results extend results of Gyárfás-Lehel-Schelp and Liu on constrained Ramsey numbers, and improve a result of Tang-Li-Yan on anti-Ramsey numbers.

Figures (2)

• Figure 1: The 3-uniform paths of length 3.
• Figure 2: The 3-graphs $S^{(3)}_2$, $S^{(3)}_3$, $\mathbb{S}^{(3)}_{2}$, $\mathbb{S}^{(3)}_{3}$ and $C^{(3)}_3$.

Theorems & Definitions (79)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
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• Theorem 1.5
• Remark 1.1
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• Theorem 1.7
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• Theorem 1.9