Bulavka, Denys ;
Goodarzi, Afshin ;
Tancer, Martin
Optimal Bounds for the Colorful Fractional Helly Theorem
Abstract
The well known fractional Helly theorem and colorful Helly theorem can be merged into the so called colorful fractional Helly theorem. It states: for every α ∈ (0, 1] and every nonnegative integer d, there is β_{col} = β_{col}(α, d) ∈ (0, 1] with the following property. Let ℱ₁, … , ℱ_{d+1} be finite nonempty families of convex sets in ℝ^d of sizes n₁, … , n_{d+1}, respectively. If at least α n₁ n₂ ⋯ n_{d+1} of the colorful (d+1)tuples have a nonempty intersection, then there is i ∈ [d+1] such that ℱ_i contains a subfamily of size at least β_{col} n_i with a nonempty intersection. (A colorful (d+1)tuple is a (d+1)tuple (F₁, … , F_{d+1}) such that F_i belongs to ℱ_i for every i.)
The colorful fractional Helly theorem was first stated and proved by Bárány, Fodor, Montejano, Oliveros, and Pór in 2014 with β_{col} = α/(d+1). In 2017 Kim proved the theorem with better function β_{col}, which in particular tends to 1 when α tends to 1. Kim also conjectured what is the optimal bound for β_{col}(α, d) and provided the upper bound example for the optimal bound. The conjectured bound coincides with the optimal bounds for the (noncolorful) fractional Helly theorem proved independently by Eckhoff and Kalai around 1984.
We verify Kim’s conjecture by extending Kalai’s approach to the colorful scenario. Moreover, we obtain optimal bounds also in a more general setting when we allow several sets of the same color.
BibTeX  Entry
@InProceedings{bulavka_et_al:LIPIcs.SoCG.2021.19,
author = {Bulavka, Denys and Goodarzi, Afshin and Tancer, Martin},
title = {{Optimal Bounds for the Colorful Fractional Helly Theorem}},
booktitle = {37th International Symposium on Computational Geometry (SoCG 2021)},
pages = {19:119:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783959771849},
ISSN = {18688969},
year = {2021},
volume = {189},
editor = {Buchin, Kevin and Colin de Verdi\`{e}re, \'{E}ric},
publisher = {Schloss Dagstuhl  LeibnizZentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/opus/volltexte/2021/13818},
URN = {urn:nbn:de:0030drops138186},
doi = {10.4230/LIPIcs.SoCG.2021.19},
annote = {Keywords: colorful fractional Helly theorem, dcollapsible, exterior algebra, drepresentable}
}
02.06.2021
Keywords: 

colorful fractional Helly theorem, dcollapsible, exterior algebra, drepresentable 
Seminar: 

37th International Symposium on Computational Geometry (SoCG 2021)

Issue date: 

2021 
Date of publication: 

02.06.2021 