# Judicious partitions of uniform hypergraphs

## TL;DR

This paper proves a new lower bound for partitioning vertices of r-uniform hypergraphs into r parts, ensuring each part meets a significant fraction of edges, improving previous bounds for r ≥ 4.

## Contribution

It establishes a weaker but improved bound for partitions of r-uniform hypergraphs, advancing understanding of hypergraph edge meet properties.

## Key findings

- For r ≥ 4, a partition into r classes exists where each meets at least (r/(3r-4))m edges.
- The result improves upon previous bounds for hypergraph partitions.
- The paper confirms the conjecture for the case r=2, and advances bounds for r ≥ 4.

## Abstract

The vertices of any graph with $m$ edges may be partitioned into two parts so that each part meets at least $\frac{2m}{3}$ edges. Bollob\'as and Thomason conjectured that the vertices of any $r$-uniform hypergraph with $m$ edges may likewise be partitioned into $r$ classes such that each part meets at least $\frac{r}{2r-1}m$ edges. In this paper we prove the weaker statement that, for each $r\ge 4$, a partition into $r$ classes may be found in which each class meets at least $\frac{r}{3r-4}m$ edges, a substantial improvement on previous bounds.