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Hall's marriage theorem

Peter J. Cameron

TL;DR

The paper provides a comprehensive treatment of Hall's marriage theorem, detailing its original statement, numerous equivalent forms, and wide-ranging connections to graph theory, posets, and matroids. It integrates classical proofs (König, Menger), algorithmic approaches (max-flow/min-cut), and counting results (permanents, Latin squares), while also exploring infinite versions and hypergraph generalizations. Key contributions include the rank-based matroid generalization, the algorithmic realization of SDR construction, and topological proofs for hypergraph extensions. The work underscores Hall's theorem as a foundational tool with deep implications across combinatorics, optimization, and design theory, shaping both theory and computation.

Abstract

In 1935, Philip Hall published what is often referred to as ``Hall's marriage theorem'' in a short paper (P.~Hall, On Representatives of Subsets, \textit{J. Lond. Math. Soc.} (1) \textbf{10} (1935), no.1, 26--30.) This paper has been very influential. I state the theorem and outline Hall's proof, together with some equivalent (or stronger) earlier results, and proceed to discuss some the many directions in combinatorics and beyond which this theorem has influenced.

Hall's marriage theorem

TL;DR

The paper provides a comprehensive treatment of Hall's marriage theorem, detailing its original statement, numerous equivalent forms, and wide-ranging connections to graph theory, posets, and matroids. It integrates classical proofs (König, Menger), algorithmic approaches (max-flow/min-cut), and counting results (permanents, Latin squares), while also exploring infinite versions and hypergraph generalizations. Key contributions include the rank-based matroid generalization, the algorithmic realization of SDR construction, and topological proofs for hypergraph extensions. The work underscores Hall's theorem as a foundational tool with deep implications across combinatorics, optimization, and design theory, shaping both theory and computation.

Abstract

In 1935, Philip Hall published what is often referred to as ``Hall's marriage theorem'' in a short paper (P.~Hall, On Representatives of Subsets, \textit{J. Lond. Math. Soc.} (1) \textbf{10} (1935), no.1, 26--30.) This paper has been very influential. I state the theorem and outline Hall's proof, together with some equivalent (or stronger) earlier results, and proceed to discuss some the many directions in combinatorics and beyond which this theorem has influenced.

Paper Structure

This paper contains 10 sections, 11 theorems, 6 equations.

Table of Contents

  1. Introduction
  2. Precursors and equivalents
  3. Dilworth's Theorem and perfect graphs
  4. Matroids
  5. Counting SDRs
  6. Latin squares
  7. Algorithmic version
  8. The infinite
  9. A final variant
  10. Acknowledgement

Key Result

Theorem 1.1

A necessary and sufficient condition for the existence of an SDR for a tuple $(T_1,\ldots,T_n)$ of sets is that, for $0\le k\le n$, any $k$ of the sets contain between them at least $k$ elements of $S$.

Theorems & Definitions (16)

  • Theorem 1.1
  • Proof 1
  • Theorem 2.1
  • Proof 2
  • Theorem 2.2
  • Proof 3
  • Theorem 2.3
  • Theorem 3.1
  • Theorem 4.1
  • Theorem 5.1
  • ...and 6 more