The topology connecting graph colorings, graph matchings, and discrete geometry

The classical Borsuk–Ulam theorem states that for any continuous map
from the sphere to Euclidean space, f: S^d → R^d, there is a pair of
antipodal points that are identified, so f(x) = f(−x). Through a
generalization of this theorem, we provide a new proof of topological
Hall's theorem that relates directly to chromatic number bounds and
Radon's theorem in discrete geometry. By tying these results together,
we can provide more structural insights to the combinatorial problems.