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Hall's Harem Theorem with controlled sizes of cycles

Karol Duda

TL;DR

This work proves a new variant of Hall's Harem Theorem for fully reflected, locally finite bipartite graphs, yielding a perfect $(1,d-1)$-matching realized by a unary function $f:\mathbb{N}\to\mathbb{N}$ with controlled cycle sizes. The authors present an inductive construction that preserves $U^{(n)}$-reflectedness and the Hall $d$-harem condition at each step, while engineering cycles so that every element eventually reaches a periodic point and cycle lengths are effectively bounded. The approach extends the classical Hall framework with the notion of reflections and a carefully orchestrated sequence of $(1,d)$-fans, enabling a controllable dynamical behavior of $f$ and establishing independence from the companion computable version arXiv:2105.06304. The results have potential implications for computable amenability contexts and related coarse-geometric constructions, by providing a classical, noncomputable route to the same cycle-control phenomenon.

Abstract

We prove a new version of Hall's Harem Theorem, where the final matching is realized by a unary function with additional conditions on behavior of cycles. The present paper can be considered as a helpful companion of the paper of the author: arXiv:2105.06304, where a computable version of Hall's Harem Theorem with controlled sizes of cycles is proved. These two versions of Hall's Harem Theorem are independent: none of them follows from the other one.

Hall's Harem Theorem with controlled sizes of cycles

TL;DR

This work proves a new variant of Hall's Harem Theorem for fully reflected, locally finite bipartite graphs, yielding a perfect -matching realized by a unary function with controlled cycle sizes. The authors present an inductive construction that preserves -reflectedness and the Hall -harem condition at each step, while engineering cycles so that every element eventually reaches a periodic point and cycle lengths are effectively bounded. The approach extends the classical Hall framework with the notion of reflections and a carefully orchestrated sequence of -fans, enabling a controllable dynamical behavior of and establishing independence from the companion computable version arXiv:2105.06304. The results have potential implications for computable amenability contexts and related coarse-geometric constructions, by providing a classical, noncomputable route to the same cycle-control phenomenon.

Abstract

We prove a new version of Hall's Harem Theorem, where the final matching is realized by a unary function with additional conditions on behavior of cycles. The present paper can be considered as a helpful companion of the paper of the author: arXiv:2105.06304, where a computable version of Hall's Harem Theorem with controlled sizes of cycles is proved. These two versions of Hall's Harem Theorem are independent: none of them follows from the other one.

Paper Structure

This paper contains 26 sections, 7 theorems, 49 equations, 10 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. General preliminaries
  4. Reflections
  5. Controlled sizes of cycles. Main theorem
  6. Notation used in the construction
  7. The construction
  8. Step 1, part 1
  9. Remarks before part 2
  10. Step 1, part 2
  11. Step 1, part 3
  12. The output of the first step
  13. The situation before step $n+1$
  14. Step n+1, part 1
  15. The output of part 1
  16. ...and 11 more sections

Key Result

Theorem 2.3

csc Let $\Gamma=(U,V,E)$ be a locally finite graph and let $k\in \mathbb{N},\; k\geq 1$. The following conditions are equivalent:

Figures (10)

  • Figure 1: The first part of the first step, $\mathfrak{M}^1_{0}$-fan of $u_0$ in red and green, $M^0_0$ in green.
  • Figure 2: Step 1, part 2. $\mathfrak{M}^2_{0}$ is red. We want the edge $(u^{1}_0, v_{u_0})$ to be in $M_0$. Force the situation from Figure 4.
  • Figure 3: $M^1_{0}$ is green. It is possible that the purple fan consisting of edges $(\dot{u}^{\perp}_0,\dot{v}^{\perp}_{0,1}),(\dot{u}^{\perp}_0,\dot{v}^{\perp}_{0,2})$ will be added to $\Gamma^{(1)\perp}$.
  • Figure 4: $\Gamma^{(n)\star}$ is black, $\Gamma^{(n)\perp}$ is purple.
  • Figure 5: $\mathfrak{M}^n_1$ is red and green, $M_n^0$ is green. We have $v_{u_n}=v^{\perp}_{j_0,2}$.
  • ...and 5 more figures

Theorems & Definitions (29)

  • Definition 2.1
  • Definition 2.2
  • Theorem 2.3: Hall's Harem theorem
  • Definition 2.4
  • Definition 2.5
  • Theorem 2.6
  • Claim 3.1
  • Remark 3.2
  • Claim 3.3
  • Claim 3.4
  • ...and 19 more