On multicolor Ramsey numbers of triple system paths of length 3

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 $\left(\genfrac{}{}{0ex}{}{[n]}{3}\right)$ 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.

18 pages, 3 figures

References in corpus (2)

• Multicolour Ramsey numbers of paths and even cycles
• Paths in hypergraphs: a rescaling phenomenon