Colorful Words and d-Tverberg Complexes

TL;DR

This paper provides a complete combinatorial characterization of weakly d-Tverberg complexes, clarifying which intersection patterns of convex hulls necessarily occur in large point sets in -dimensional space, and constructs specific graphs with particular Tverberg properties.

Contribution

It offers a full combinatorial description of weakly d-Tverberg complexes and constructs graphs with unique Tverberg intersection properties, answering an open question.

Findings

Characterization of weakly d-Tverberg complexes

Construction of graphs not weakly d'-Tverberg for any d' 

Strengthening the concept of d-representable complexes

Abstract

We give a complete combinatorial characterization of weakly $d$-Tverberg complexes. These complexes record which intersection combinatorics of convex hulls necessarily arise in any sufficiently large general position point set in $\mathbb R^d$. This strengthens the concept of $d$-representable complexes, which describe intersection combinatorics that arise in at least one point set. Our characterization allows us to construct for every fixed $d$ a graph that is not weakly $d'$-Tverberg for any $d'\le d$, answering a question of De Loera, Hogan, Oliveros, and Yang.