Tverberg-Type Theorems for Matroids: A Counterexample and a Proof
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Bárány, 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.
Mathematics Subject Classification (2010)52A35 05B35
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We thank the referees of Combinatorica for very detailed and helpful comments, including in particular a simplification for the proof of Theorem 1.3.
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