On multicolor Ramsey number for 3-paths of length three

TL;DR

This paper establishes a new upper bound on the multicolor Ramsey number for 3-paths of length three in 3-uniform hypergraphs, demonstrating that a sufficiently large complete hypergraph guarantees a monochromatic loose path of length three.

Contribution

The paper provides a novel upper bound on the multicolor Ramsey number for 3-paths of length three in 3-uniform hypergraphs, advancing understanding of hypergraph colorings.

Findings

If hyperedges of a complete 3-uniform hypergraph on 2n+√(18n+1)+2 vertices are n-colored, a monochromatic 3-path of length three exists.

The result improves bounds on multicolor Ramsey numbers for specific hypergraph paths.

The proof involves combinatorial and coloring arguments specific to 3-uniform hypergraphs.

Abstract

We show that if we color the hyperedges of the complete $3$-uniform complete graph on $2n+\sqrt{18n+1}+2$ vertices with $n$ colors, then one of the color classes contains a loose path of length three.