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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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A geometric Hall-type theorem
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by Andreas F. Holmsen, Leonardo Martinez-Sandoval and Luis Montejano PDF
Proc. Amer. Math. Soc. 144 (2016), 503-511 Request permission

Abstract:

We introduce a geometric generalization of Hall’s marriage theorem. For any family $F = \{X_1, \dots , X_m\}$ of finite sets in $\mathbb {R}^d$, we give conditions under which it is possible to choose a point $x_i\in X_i$ for every $1\leq i \leq m$ in such a way that the points $\{x_1,\dots ,x_m\}\subset \mathbb {R}^d$ 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.
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Additional Information
  • Andreas F. Holmsen
  • Affiliation: Department of Mathematical Sciences, KAIST, Daejeon, South Korea
  • MR Author ID: 685253
  • Email: andreash@kaist.edu
  • Leonardo Martinez-Sandoval
  • Affiliation: Instituto de Matemáticas, National University of Mexico at Querétaro, Juriquilla, Querétaro 76230, Mexico – and – Institut de Mathémathiques et de Modélisation de Montpellier, Univesité de Montpellier, Place Eugéne Bataillon, 34095 Montpellier Cedex, France
  • MR Author ID: 1004558
  • Email: leomtz@im.unam.mx
  • Luis Montejano
  • Affiliation: Instituto de Matemáticas, National University of Mexico at Querétaro, Juriquilla , Querétaro 76230, Mexico
  • MR Author ID: 126505
  • Email: luis@matem.unam.mx
  • Received by editor(s): December 20, 2014
  • Received by editor(s) in revised form: January 8, 2015, and January 14, 2015
  • Published electronically: June 26, 2015
  • Additional Notes: The first author would like to thank the Instituto de Matemáticas, UNAM at Querétaro for their hospitality and support during his visit. The second and third authors wish to acknowledge support from CONACyT under Project 166306, support from PAPIIT–UNAM under Project IN112614 and support from ECOS Nord project M13M01. The third author was supported by CONACyT Scholarship 277462
  • Communicated by: Patricia L. Hersh
  • © Copyright 2015 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 144 (2016), 503-511
  • MSC (2010): Primary 05D15, 52C35
  • DOI: https://doi.org/10.1090/proc12733
  • MathSciNet review: 3430829