A geometric hall-type theorem

Andreas F. Holmsen; Leonardo Martinez-Sandoval; Luis Montejano

doi:10.1090/proc12733

A geometric hall-type theorem

Andreas F. Holmsen, Leonardo Martinez-Sandoval, Luis Montejano

Research output: Contribution to journal › Article › peer-review

5 Scopus citations

Abstract

We introduce a geometric generalization of Hall’s marriage theorem. For any family F = {X₁,..., X_m} of finite sets in ℝ^d, we give conditions under which it is possible to choose a point x_i ∈ X_i for every 1 ≤ i ≤ m in such a way that the points {x₁,..., x_m} ⊂ ℝ^d are in general position. We give two proofs, one elementary proof requiring slightly stronger conditions, and one proof using topological techniques in the spirit of Aharoni and Haxell’s celebrated generalization of Hall’s theorem.

Original language	English
Pages (from-to)	503-511
Number of pages	9
Journal	Proceedings of the American Mathematical Society
Volume	144
Issue number	2
DOIs	https://doi.org/10.1090/proc12733
State	Published - 1 Feb 2016
Externally published	Yes

Keywords

Hall’s theorem
Matroids
Points in general position
Topological combinatorics

ASJC Scopus subject areas

General Mathematics
Applied Mathematics

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10.1090/proc12733

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title = "A geometric hall-type theorem",

abstract = "We introduce a geometric generalization of Hall{\textquoteright}s marriage theorem. For any family F = {X1,..., Xm} of finite sets in ℝd, we give conditions under which it is possible to choose a point xi ∈ Xi for every 1 ≤ i ≤ m in such a way that the points {x1,..., xm} ⊂ ℝd are in general position. We give two proofs, one elementary proof requiring slightly stronger conditions, and one proof using topological techniques in the spirit of Aharoni and Haxell{\textquoteright}s celebrated generalization of Hall{\textquoteright}s theorem.",

keywords = "Hall{\textquoteright}s theorem, Matroids, Points in general position, Topological combinatorics",

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T1 - A geometric hall-type theorem

AU - Holmsen, Andreas F.

AU - Martinez-Sandoval, Leonardo

AU - Montejano, Luis

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N2 - We introduce a geometric generalization of Hall’s marriage theorem. For any family F = {X1,..., Xm} of finite sets in ℝd, we give conditions under which it is possible to choose a point xi ∈ Xi for every 1 ≤ i ≤ m in such a way that the points {x1,..., xm} ⊂ ℝd are in general position. We give two proofs, one elementary proof requiring slightly stronger conditions, and one proof using topological techniques in the spirit of Aharoni and Haxell’s celebrated generalization of Hall’s theorem.

AB - We introduce a geometric generalization of Hall’s marriage theorem. For any family F = {X1,..., Xm} of finite sets in ℝd, we give conditions under which it is possible to choose a point xi ∈ Xi for every 1 ≤ i ≤ m in such a way that the points {x1,..., xm} ⊂ ℝd are in general position. We give two proofs, one elementary proof requiring slightly stronger conditions, and one proof using topological techniques in the spirit of Aharoni and Haxell’s celebrated generalization of Hall’s theorem.

KW - Hall’s theorem

KW - Matroids

KW - Points in general position

KW - Topological combinatorics

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JO - Proceedings of the American Mathematical Society

JF - Proceedings of the American Mathematical Society

IS - 2

ER -

A geometric hall-type theorem

Abstract

Keywords

ASJC Scopus subject areas

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