  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.
