{
  "id": "1705.03624",
  "version": "v1",
  "published": "2017-05-10T06:49:19.000Z",
  "updated": "2017-05-10T06:49:19.000Z",
  "title": "Tverberg-type theorems for matroids: A counterexample and a proof",
  "authors": [
    "Pavle V. M. Blagojević",
    "Albert Haase",
    "Günter M. Ziegler"
  ],
  "comment": "16 pages, 4 figures",
  "categories": [
    "math.CO",
    "math.AT",
    "math.MG"
  ],
  "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.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-05-10T06:49:19.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05E45",
      "52A35"
    ],
    "keywords": [
      "counterexample",
      "topological radon theorem",
      "disjoint bases",
      "tight tverberg-type theorem",
      "guarantees multiple coincidences"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 16,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}