Structure and Decomposition of Deltoids in Abelian Groups

TL;DR

This paper explores the structure and conditions for partial matchings in abelian groups, introducing the concept of deltoids, and provides new theoretical results on minimal defects and obstructions.

Contribution

It develops a comprehensive framework for understanding defective matchings in abelian groups, including necessary and sufficient conditions and structural obstructions.

Findings

Characterization of when partial matchings exist with prescribed defects

Identification of minimal unavoidable defects in pairs of subsets

Max-min results on partitioning sets into admissible subsets

Abstract

Deltoids provide a natural framework for studying defective (partial) matchings in abelian groups, and we develop both structure and existence results in this setting. Given finite subsets $A$ and $B$ of an abelian group $G$, a matching is a bijection $f:A\to B$ such that $af(a)\notin A$ for all $a\in A$, a definition motivated by the study of canonical forms for symmetric tensors. We provide necessary and sufficient conditions for the existence of a partial matching with any prescribed defect, and then describe the minimal unavoidable defect for a pair $(A,B)$. We also define and examine a defective version of Chowla sets in the matching context. We prove a structure theorem identifying obstructions to the existence of partial matchings with small defect. Finally, within the deltoid setup, we establish max-min results on the partitioning of $A$ and $B$ into left- and right-admissible sets. Our tools mix results from transversal theory with ideas from additive number theory.