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We give a complete combinatorial characterization of weakly $d$-Tverbergcomplexes. These complexes record which intersection combinatorics of convexhulls 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 isnot weakly $d'$-Tverberg for any $d'\le d$, answering a question of De Loera,Hogan, Oliveros, and Yang.

Colorful Words and d-Tverberg Complexes