A geometric Hall-type theorem

Andreas Holmsen, Leonardo Martinez-Sandoval, Luis Montejano

Published 2014-12-20Version 1

We introduce a geometric generalization of Hall's marriage theorem. Given a family $F = \{X_1, \dots, X_m\}$ of finite sets in $\mathbb{R}^d$, we give conditions under which it is possible to chose a point $x_i\in X_i$, in such a way that the points $\{x_1,...,x_m\}\subset \mathbb{R}^d$ are in general position. The proof uses topological techniques in the spirit of Aharoni and Haxell's celebrated generalization of Hall's theorem.