2026/06/10 by Mohsen Aliabadi, Jozsef Losonczy
Mathematics · #math.CO
We formulate and prove linear counterparts of results on partial matchings between finite subsets in abelian groups. The chosen setting is a field extension K\subsetneq L, where we introduce a notion of partial matching between finite-dimensional K-subspaces A,B⊆ L. Our main results include (1) a characterization of those pairs (A,B) that are partially matchable up to a specified defect, (2) a decomposition theorem for pairs (A,B) having positive deficiency, and (3) an existence criterion for pairs having a prescribed dimension and satisfying a deficiency bound. We use these results to recover and extend various parts of this area of matching theory, emphasizing the close analogy between the group-theoretic and linear perspectives. Our approach blends algebraic techniques with tools from matroidal transversal theory, and utilizes a linearized version of the e-transform from additive number theory.