On a topological version of Pach's overlap theorem

Bukh, Boris; Hubard, Alfredo

doi:10.1112/blms.12302

Cited by 4 publications

(3 citation statements)

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“…Subsequent work. Considering our work, Bukh and Hubard [5] very recently improved the bound on τ (d, n) to τ (d, n) ≤ 30(ln n) 1/(d−1) . Let L be a graded lattice of rank rk( 1) = d + 1.…”

Section: Introductionmentioning

confidence: 83%

Pach’s Selection Theorem Does Not Admit a Topological Extension

Bárány

Meshulam

Nevo

et al. 2018

Discrete Comput Geom

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Let U 1 , . . . , U d+1 be n-element sets in R d . Pach's selection theorem says that there exist subsets Z 1 ⊂ U 1 , . . . , Z d+1 ⊂ U d+1 and a point u ∈ R d such that each |Z i | ≥ c 1 (d)n and u ∈ conv{z 1 , . . . , z d+1 } for every choice of z 1 ∈ Z 1 , . . . , z d+1 ∈ Z d+1 . Here we show that this theorem does not admit a topological extension with linear size sets Z i . However, there is a topological extension where each |Z i | is of order (log n) 1/d .

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Section: Introductionmentioning

confidence: 83%

Pach’s Selection Theorem Does Not Admit a Topological Extension

Bárány

Meshulam

Nevo

et al. 2018

Discrete Comput Geom

View full text Add to dashboard Cite

show abstract

“…In higher dimensions Tverberg's theorem and the colorful Carathéodory theorem imply (see [Bár82]) the following result. We mention that Pach's theorem does not have a topological extension, as shown in [BMNT18] and in [BH20] in a stonger form. 7.5.…”

Section: Radon Partitions and Radon Points For Configurations Based O...mentioning

confidence: 89%

Untitled

2022

Bull. Amer. Math. Soc.

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In this paper we present a variety of problems in the interface between combinatorics and geometry around the theorems of Helly, Radon, Carathéodory, and Tverberg. Through these problems we describe the fascinating area of Helly-type theorems and explain some of their main themes and goals. Helly, Carathéodory, and Radon theoremsIn this paper, we present a variety of problems in the interface between combinatorics and geometry around the theorems of Helly, Radon, Carathéodory, and Tverberg.Helly's theorem [Hel23] asserts that for a family {K 1 , K 2 , . . . , K n } of convex sets in R d , where n ≥ d + 1, if every d + 1 of the sets have a point in common, then all of the sets have a point in common. The closely related Carathéodory theorem [Car07] states that forThe more general colorful Carathéodory theorem [Bár82] says the following. Let S 1 , S 2 , . . . , S d+1 be d + 1 sets (or colors if you wish) in R d . Suppose that x ∈ d+1 i=1 conv S i . Then there is a transversal T = {x 1 , . . . , x d+1 } of the system S 1 , . . . , S d+1 , meaning that x 1 ∈ S 1 , x 2 ∈ S 2 , . . . , x d+1 ∈ S d+1 such that x ∈ conv T . A transversal is also called a rainbow set when S 1 , . . . , S d+1 are considered as colors. The uncolored version, that is, when S 1 = S 2 = • • • = S d+1 , is the classic result of Carathéodory. There is a closely related colorful version of Helly's theorem due to Lovász that appeared in [Bár82].Tverberg's theorem [Tve66] states the following. Let x 1 , x 2 , . . . , x m be points in R d with m ≥ (r − 1)(d + 1) + 1. Then there is a partition S 1 , S 2 , . . . , S r of {1, 2, . . . , m} such that r j=1 conv {x i : i ∈ S j } = ∅. This was a conjecture by Birch, who also proved the planar case in a slightly different form. The bound of (r −1)(d+1)+1 in the theorem is sharp as can easily be seen from the configuration of points in a sufficiently general position.The case r = 2 is Radon's theorem [Rad21], another classic from 1921, which was used by Radon to prove Helly's theorem. Helly's original proof (published later) was based on a separation argument. Sarkaria [Sar92] gave a simple proof of Tverberg's theorem based on the colorful Carathéodory theorem.

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“…We mention that Pach's theorem does not have a topological extension, as shown in [BMNT18], and in [BH20] in a stonger form.…”

Section: The Covering Number Theoremmentioning

confidence: 89%