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Short proof of the hypergraph container theorem

Part of: Graph theory

Published online by Cambridge University Press:  16 May 2025

Rajko Nenadov  and
Huy Tuan Pham
Rajko Nenadov
Affiliation:
School of Computer Science, University of Auckland, Auckland, New Zealand
Huy Tuan Pham*
Affiliation:
School of Mathematics, Institute for Advanced Study, Princeton, NJ, USA
*
Corresponding author: Huy Tuan Pham; Email: htpham@caltech.edu
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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.

Keywords

Hypergraph containersindependent sets

MSC classification

Primary: 05C30: Enumeration in graph theory 05C65: Hypergraphs 05C69: Dominating sets, independent sets, cliques
Secondary: 05C35: Extremal problems

Information

Type
Paper
Information
Combinatorics, Probability and Computing , Volume 34 , Issue 5 , September 2025 , pp. 621 - 624
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Copyright
© The Author(s), 2025. Published by Cambridge University Press

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References

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