A geometric Hall-type theorem

TL;DR

This paper extends Hall's marriage theorem into a geometric context, providing conditions for selecting points from finite sets in Euclidean space so that they are in general position, with proofs using elementary and topological methods.

Contribution

It introduces a geometric generalization of Hall's theorem for finite sets in Euclidean space, with two different proof techniques.

Findings

Established conditions for selecting points in general position.

Provided an elementary proof with stronger conditions.

Developed a topological proof inspired by Aharoni and Haxell.

Abstract

We introduce a geometric generalization of Hall's marriage theorem. For any family $F = \{X_1, \dots, X_m\}$ of finite sets in $\mathbb{R}^d$, we give conditions under which it is possible to choose a point $x_i\in X_i$ for every $1\leq i \leq m$ in such a way that the points $\{x_1,...,x_m\}\subset \mathbb{R}^d$ are in general position. We give two proofs, one elementary proof requiring slightly stronger conditions, and one proof using topological techniques in the spirit of Aharoni and Haxell's celebrated generalization of Hall's theorem.