Colorful Words and d-Tverberg Complexes

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.