Tverberg-Type Theorems with Trees and Cycles as (Nerve) Intersection Patterns

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 but a special case of a more general situation. Given sufficiently many points, all trees and cycles can also be induced by at least one partition of a point set.