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Combinatorica

, Volume 39, Issue 3, pp 477–500 | Cite as

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

  • Pavle V. M. Blagojević
  • Albert Haase
  • Günter M. ZieglerEmail author
Original Paper
  • 31 Downloads

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.

Mathematics Subject Classification (2010)

52A35 05B35 

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Notes

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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Copyright information

© János Bolyai Mathematical Society and Springer-Verlag Berlin Heidelberg 2019

Authors and Affiliations

  • Pavle V. M. Blagojević
    • 1
    • 2
  • Albert Haase
    • 1
  • Günter M. Ziegler
    • 1
    Email author
  1. 1.Inst. Math.FU BerlinBerlinGermany
  2. 2.Mat. Institut SANUBeogradSerbia

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