Tverberg-Type Theorems with Altered Intersection Patterns (Nerves)

Abstract

Tverberg’s theorem says that a set with sufficiently many points in $${\mathbb {R}}^d$$ can always be partitioned into m parts so that the $$(m-1)$$ -simplex is the (nerve) intersection pattern of the convex hulls of the parts. The main results of our paper demonstrate that Tverberg’s theorem is just a special case of a more general situation, where other simplicial complexes must always arise as nerve complexes, as soon as the number of points is large enough. We prove that, given a set with sufficiently many points, all trees and all cycles can also be induced by at least one partition of the point set. We also discuss how some simplicial complexes can never be achieved this way, even for arbitrarily large sets of points.