Covering and labeling generalizations of the Borsuk--Ulam theorem

The classical Borsuk--Ulam theorem states that for any continuous map from the d-sphere to d-dimensional Euclidean space, there is a pair of antipodal points that are identified.

We generalize Fan's combinatorial labeling result for sphere triangulations, yielding a comprehensive extension of the Borsuk--Ulam theorem. In typical applications, the Borsuk--Ulam theorem asserts the existence of some object, whereas our extension uses finite point sets in Euclidean space to additionally provide insight into the structure of those objects. We develop some of those applications for sphere coverings, for example Kneser-type colorings, Hall-type results for hypergraphs, hyperplane mass partitions, and topological lower bounds for chromatic numbers.