---
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_language: eng
dct_publisher: Springer@
dct_title: Bounds for Pach's selection theorem and for the minimum solid angle in
a simplex@
...