3-uniform monotone paths and multicolor Ramsey numbers

TL;DR

This paper establishes bounds relating multicolor Ramsey numbers for triangles to hypergraph Ramsey numbers involving monotone paths, connecting a longstanding open problem to exponential growth questions.

Contribution

It introduces bounds linking multicolor Ramsey numbers for triangles with hypergraph Ramsey numbers for monotone paths, providing a new perspective on an old problem.

Findings

Proves that $r(3;n) \,\leq\, R(P_{n+2},\mathcal{J}_n)$

Establishes an upper bound $R(P_{n+2},\mathcal{J}_n) \leq 4^n \cdot r(3;n)$

Shows the equivalence of exponential growth between $r(3;n)$ and $R(P_{n+2},\mathcal{J}_n)$.

Abstract

The monotone path $P_{n+2}$ is an ordered 3-uniform hypergraph whose vertex set has size $n+2$ and edge set consists of all consecutive triples. In this note, we consider the collection $\mathcal{J}_n$ of ordered 3-uniform hypergraphs named monotone paths with $n$ jumps, and we prove the following relation \begin{equation*} r(3;n) \leq R(P_{n+2},\mathcal{J}_n) \leq 4^n \cdot r(3;n), \end{equation*} where $r(3;n)$ is the multicolor Ramsey number for triangles and $R(P_{n+2},\mathcal{J}_n)$ is the hypergraph Ramsey number for $P_{n+2}$ versus any member of $\mathcal{J}_n$. In particular, whether $r(3;n)$ is exponential, which is a very old problem of Erd\H{o}s, is equivalent to whether $R(P_{n+2},\mathcal{J}_n)$ is exponential.