The topological Tverberg problem beyond prime powers

Frick, Florian; Soberón, Pablo

doi:10.48550/arxiv.2005.05251

Cited by 4 publications

(4 citation statements)

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“…The main difficulty is how to use the weak general position condition. A recent result of Frick and Soberón [FS20] (see Section 7.1) is perhaps relevant here. While the conclusion of the cascade conjecture seems stronger than that of Reay's dimension conjecture, it is not known how to derive it from the cascade conjecture.…”

Section: Conjecture 52 (Reay's Conjecture) Ifmentioning

confidence: 80%

“…A recent result of Frick and Soberón [FS20] is that a set of r(d + 1) points in R d can always be partitioned into r sets, each of size d + 1, such that the convex hulls of the parts have a point in common. This theorem is related to the uncolored case of Problem 7.1 but without the word "interior".…”

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

confidence: 99%

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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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Section: Conjecture 52 (Reay's Conjecture) Ifmentioning

confidence: 80%

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

confidence: 99%

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2022

Bull. Amer. Math. Soc.

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“…[BSS81]). Also a generalization of Birch's Theorem, a weaker version of Tverberg's Theorem, was recently proven for topological drawings of complete graphs [FS20]. The general case, however, remains unknown.…”

Section: Discussionmentioning

confidence: 99%

Topological Drawings meet Classical Theorems from Convex Geometry

Bergold¹,

Felsner²,

Scheucher³

et al. 2020

Preprint

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In this article we discuss classical theorems from Convex Geometry in the context of topological drawings. In a simple topological drawing of the complete graph Kn, any two edges share at most one point: either a common vertex or a point where they cross. Triangles of simple topological drawings can be viewed as convex sets, this gives a link to convex geometry.We present a generalization of Kirchberger's Theorem, a family of simple topological drawings with arbitrarily large Helly number, and a new proof of a topological generalization of Carathéodory's Theorem in the plane. We also discuss further classical theorems from Convex Geometry in the context of simple topological drawings.We introduce "generalized signotopes" as a generalization of topological drawings. As indicated by the name they are a generalization of signotopes, a structure studied in the context of encodings for arrangements of pseudolines.

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“…(E.g. compare [Sk16, §1] citing [BZ16] to [BZ16, §1] not citing [Sk16], and [Sh18] citing [Sk16] to [BS17], [FS20] not citing [Sk16].) Remark 1.2.…”

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confidence: 99%

On different reliability standards in current mathematical research

Skopenkov¹

2021

Preprint

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In this note I expose the reliability standards I share, and give examples of different reliability standards. Contents1. On different reliability standards 1 2. Reviewing peer review system 7 References 91. On different reliability standards By 'reliability' I understand validity and novelty of results. This is closely related, but not the same, as clarity. This is independent of value and importance of results.Reliability standards in mathematics change. E.g. a long time ago, oral proof was considered as reliable. As proofs gradually became more complicated, there appeared the understanding (and then the rule) that only written proof can be considered to be reliable or not (and so could constitute a fair claim for a result).

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The topological Tverberg problem beyond prime powers

Cited by 4 publications

References 9 publications

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Topological Drawings meet Classical Theorems from Convex Geometry

On different reliability standards in current mathematical research

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