---
res:
  bibo_abstract:
  - 'We estimate the selection constant in the following geometric selection theorem
    by Pach: For every positive integer d, there is a constant (Formula presented.)
    such that whenever (Formula presented.) are n-element subsets of (Formula presented.),
    we can find a point (Formula presented.) and subsets (Formula presented.) for
    every i∈[d+1], each of size at least cdn, such that p belongs to all rainbowd-simplices
    determined by (Formula presented.) simplices with one vertex in each Yi. We show
    a super-exponentially decreasing upper bound (Formula presented.). The ideas used
    in the proof of the upper bound also help us to prove Pach’s theorem with (Formula
    presented.), which is a lower bound doubly exponentially decreasing in d (up to
    some polynomial in the exponent). For comparison, Pach’s original approach yields
    a triply exponentially decreasing lower bound. On the other hand, Fox, Pach, and
    Suk recently obtained a hypergraph density result implying a proof of Pach’s theorem
    with (Formula presented.). In our construction for the upper bound, we use the
    fact that the minimum solid angle of every d-simplex is super-exponentially small.
    This fact was previously unknown and might be of independent interest. For the
    lower bound, we improve the ‘separation’ part of the argument by showing that
    in one of the key steps only d+1 separations are necessary, compared to 2d separations
    in the original proof. We also provide a measure version of Pach’s theorem.@eng'
  bibo_authorlist:
  - foaf_Person:
      foaf_givenName: Roman
      foaf_name: Karasev, Roman
      foaf_surname: Karasev
  - foaf_Person:
      foaf_givenName: Jan
      foaf_name: Kynčl, Jan
      foaf_surname: Kynčl
  - foaf_Person:
      foaf_givenName: Pavel
      foaf_name: Paták, Pavel
      foaf_surname: Paták
  - foaf_Person:
      foaf_givenName: Zuzana
      foaf_name: Patakova, Zuzana
      foaf_surname: Patakova
    orcid: 0000-0002-3975-1683
  - foaf_Person:
      foaf_givenName: Martin
      foaf_name: Tancer, Martin
      foaf_surname: Tancer
      foaf_workInfoHomepage: http://www.librecat.org/personId=38AC689C-F248-11E8-B48F-1D18A9856A87
    orcid: 0000-0002-1191-6714
  bibo_doi: 10.1007/s00454-015-9720-z
  bibo_issue: '3'
  bibo_volume: 54
  dct_date: 2015^xs_gYear
  dct_identifier:
  - UT:000360702400004
  dct_language: eng
  dct_publisher: Springer@
  dct_title: Bounds for Pach's selection theorem and for the minimum solid angle in
    a simplex@
...