On multicolor Ramsey numbers of triple system paths of length 3

TL;DR

This paper establishes upper bounds on the multicolor Ramsey numbers for two specific 3-uniform hypergraph paths, showing they grow linearly with the number of colors, which advances understanding in hypergraph Ramsey theory.

Contribution

The paper provides new linear upper bounds for the multicolor Ramsey numbers of loose and messy 3-uniform paths, improving previous bounds for large numbers of colors.

Findings

r_k(ℒ) < 1.54k for large k

r_k(ℳ) < 1.6k for large k

linear growth bounds established

Abstract

Let $\mathcal{H}$ be a 3-uniform hypergraph. The multicolor Ramsey number $ r_k(\mathcal{H})$ is the smallest integer $n$ such that every coloring of $ \binom{[n]}{3}$ with $k$ colors has a monochromatic copy of $\mathcal{H}$. Let $ \mathcal{L}$ be the loose 3-uniform path with 3 edges and $ \mathcal{M}$ denote the messy 3-uniform path with 3 edges; that is, let $\mathcal{L} = \{abc, cde, efg\}$ and $\mathcal{M} = \{ abc, bcd, def\}$. In this note we prove $ r_k(\mathcal{L}) < 1.54k$ and $ r_k(\mathcal{M}) < 1.6k$ for $k$ sufficiently large.