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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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Intersection patterns of finite sets and of convex sets
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by Florian Frick PDF
Proc. Amer. Math. Soc. 145 (2017), 2827-2842 Request permission

Abstract:

The main result is a common generalization of results on lower bounds for the chromatic number of $r$-uniform hypergraphs and some of the major theorems in Tverberg-type theory, which is concerned with the intersection pattern of faces in a simplicial complex when continuously mapped to Euclidean space. As an application we get a simple proof of a generalization of a result of Kriz for certain parameters. This specializes to a short and simple proof of Kneser’s conjecture. Moreover, combining this result with recent work of Mabillard and Wagner we show that the existence of certain equivariant maps yields lower bounds for chromatic numbers. We obtain an essentially elementary proof of the result of Schrijver on the chromatic number of stable Kneser graphs. In fact, we show that every neighborly even-dimensional polytope yields a small induced subgraph of the Kneser graph of the same chromatic number. We furthermore use this geometric viewpoint to give tight lower bounds for the chromatic number of certain small subhypergraphs of Kneser hypergraphs.
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Additional Information
  • Florian Frick
  • Affiliation: Department of Mathematics, Cornell University, Ithaca, New York 14853
  • MR Author ID: 1079440
  • ORCID: 0000-0002-7635-744X
  • Email: ff238@cornell.edu
  • Received by editor(s): July 9, 2016
  • Received by editor(s) in revised form: August 14, 2016, and August 15, 2016
  • Published electronically: December 30, 2016
  • Communicated by: Patricia L. Hersh
  • © Copyright 2016 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 145 (2017), 2827-2842
  • MSC (2010): Primary 05C15, 52A35
  • DOI: https://doi.org/10.1090/proc/13443
  • MathSciNet review: 3637933