Geometric variants of Hallʼs Theorem through Spernerʼs Lemma

Hallʼs marriage theorem can be regarded as a condition for the existence of a rainbow set of distinct points. Motivated by this interpretation, we introduce the notion of geometric Hall-type problems. We prove that a linear Hall-type condition guarantees the existence of a rainbow set of pairwise disjoint unit balls in Rn. We also prove that a quadratic Hall-type condition guarantees the existence of a rainbow set of points in general position on the plane. To prove these results, we present and extend a topological technique by Aharoni and Haxell that uses Spernerʼs lemma in tight triangulations of the k-dimensional simplex.