Tverberg-Type Theorems for Matroids: A Counterexample and a Proof

Barany, Kalai, and Meshulam recently obtained a topological Tverberg-type theorem for matroids, which guarantees multiple coincidences for continuous maps from a matroid complex into ℝd, if the matroid has sufficiently many disjoint bases. They make a conjecture on the connectivity of k-fold deleted joins of a matroid with many disjoint bases, which would yield a much tighter result — but we provide a counterexample already for the case of k = 2, where a tight Tverberg-type theorem would be a topological Radon theorem for matroids. Nevertheless, we prove the topological Radon theorem for the counterexample family of matroids by an index calculation, despite the failure of the connectivity-based approach.