Matchings in Matroids over Abelian Groups, III

Abstract: In an abelian group $G$, a \emph{matching} is a bijection $f\colon A\to B$ between finite subsets $A,B\subseteq G$ such that $a+f(a)\notin A$ for all $a\in A$. We say that $G$ has the \emph{matching property} if every pair of finite subsets $A,B\subseteq G$ with $|A|=|B|$ and $0\notin B$ admits such a matching. This paper develops matroidal analogues of classical results on group matchings. By embedding matroid ground sets in $G$, we introduce base matchings between matroid bases, recovering the group-theoretic setting in the uniform case, and derive structural and combinatorial criteria for their existence. Our methods blend techniques from matroid theory, group theory, and additive number theory. Our main focus is on paving matroids, a class conjectured to constitute asymptotically almost all matroids. We prove symmetric self-matchability for all paving matroids, extend asymmetric results via the hyperplane-nullity parameter, and connect stressed hyperplanes to matchability through relaxation, bridging paving and uniform matroids. This paper continues a line of research initiated in [3,4], yet is written to be self-contained and may be read independently.