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.

Mathematics Subject Classifications: 52A35, 52C45

Keywords: Tverberg's theorem, word representable, \(d\)-representable, nerve, general position, strong general position, fully independent