# On multicolor Ramsey numbers for loose $k$-paths of length three

## TL;DR

This paper proves that for any number of colors and sufficiently large hypergraph size, one color must contain a loose path of length three, establishing a bound related to multicolor Ramsey numbers for hypergraphs.

## Contribution

It establishes an upper bound on multicolor Ramsey numbers for loose paths of length three in complete k-uniform hypergraphs, showing existence of a universal constant.

## Key findings

- Existence of a universal constant A for the bound.
- For large enough hypergraphs, a monochromatic loose path of length three always exists.
- The result applies to all k ≥ 2 and any number of colors r.

## Abstract

We show that there exists an absolute constant $A$ such that for each $k\ge2$ and every coloring of the edges of the complete $k$-uniform hypergraph on $ Ar$ vertices with $r$ colors, one of the color classes contains a loose path of length three.