# Tverberg-type theorems for matroids: A counterexample and a proof

## TL;DR

This paper investigates Tverberg-type theorems for matroids, providing a counterexample to a conjecture on connectivity and proving a topological Radon theorem for that case through index calculations.

## Contribution

It presents a counterexample to a conjecture on matroid connectivity and establishes a topological Radon theorem for that specific family of matroids.

## Key findings

- Counterexample disproves the conjecture for k=2
- Topological Radon theorem holds for the counterexample matroids
- Connectivity-based approach fails, but index calculations succeed

## Abstract

B\'ar\'any, Kalai, and Meshulam recently obtained a topological Tverberg-type theorem for matroids, which guarantees multiple coincidences for continuous maps from a matroid complex to d-dimensional Euclidean space, 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.