Hall's Harem Theorem with controlled sizes of cycles

TL;DR

This paper presents a new version of Hall's Harem Theorem involving a unary function with cycle behavior constraints, complementing a previous computable version with independent proofs.

Contribution

It introduces a novel variant of Hall's Harem Theorem with cycle control, expanding the theoretical framework of the original theorem.

Findings

Proves a new version of Hall's Harem Theorem with cycle behavior conditions.

Establishes independence from a prior computable version of the theorem.

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.