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

Abstract

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.

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Acknowledgement

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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Correspondence to Günter M. Ziegler.

Additional information

P. V. M. B. received funding from DFG via the Collaborative Research Center TRR 109 “Discretization in Geometry and Dynamics” and the grant ON 174024 of the Serbian Ministry of Education and Science.

A. H. was supported by DFG via the Berlin Mathematical School.

G. M. Z. received funding from DFG via the Collaborative Research Center TRR 109 “Discretization in Geometry and Dynamics”.

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Blagojević, P.V.M., Haase, A. & Ziegler, G.M. Tverberg-Type Theorems for Matroids: A Counterexample and a Proof. Combinatorica 39, 477–500 (2019). https://doi.org/10.1007/s00493-018-3846-6

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Mathematics Subject Classification (2010)

  • 52A35
  • 05B35