# Partitioning infinite hypergraphs into few monochromatic Berge-paths

## TL;DR

This paper extends Rado's result to hypergraphs, showing that infinite complete hypergraphs can be partitioned into a limited number of monochromatic Berge-paths of different colors, with a construction proving optimality.

## Contribution

It generalizes Rado's theorem to hypergraphs, establishing optimal bounds for partitioning infinite hypergraphs into monochromatic Berge-paths.

## Key findings

- Partition of infinite hypergraphs into monochromatic Berge-paths is possible within the established bounds.
- The bounds for the number of paths are proven to be optimal through a specific construction.
- The result extends classical graph Ramsey theory to hypergraph settings.

## Abstract

Extending a result of Rado to hypergraphs, we prove that for all $s, k, t \in \mathbb{N}$ with $k \geq t \geq 2$, the vertices of every $r = s(k-t+1)$-edge-coloured countably infinite complete $k$-graph can be partitioned into the cores of at most $s$ monochromatic $t$-tight Berge-paths of different colours. We further describe a construction showing that this result is best possible.