Short proof of the hypergraph container theorem

TL;DR

This paper provides a concise and simplified proof of the hypergraph container theorem by modifying the original algorithm to improve analysis clarity.

Contribution

It introduces a new, streamlined proof of the hypergraph container theorem using a modified iterative algorithm that simplifies the analysis.

Findings

Simplified proof of the hypergraph container theorem.

Modified algorithm postpones removal of high-degree vertices.

Enhanced understanding of independent set structure in hypergraphs.

Abstract

We present a short and simple proof of the celebrated hypergraph container theorem of Balogh--Morris--Samotij and Saxton--Thomason. On a high level, our argument utilises the idea of iteratively taking vertices of largest degree from an independent set and constructing a hypergraph of lower uniformity which preserves independent sets and inherits edge distribution. The original algorithms for constructing containers also remove in each step vertices of high degree which are not in the independent set. Our modified algorithm postpones this until the end, which surprisingly results in a significantly simplified analysis.