Colorful Coverings of Polytopes and Piercing Numbers of Colorful d-Intervals

Abstract

We prove a common strengthening of Bárány’s colorful Carathéodory theorem and the KKMS theorem. In fact, our main result is a colorful polytopal KKMS theorem, which extends a colorful KKMS theorem due to Shih and Lee [Math. Ann. 296 (1993), no. 1, 35–61] as well as a polytopal KKMS theorem due to Komiya [Econ. Theory 4 (1994), no. 3, 463–466]. The (seemingly unrelated) colorful Carathéodory theorem is a special case as well. We apply our theorem to establish an upper bound on the piercing number of colorful d-interval hypergraphs, extending earlier results of Tardos [Combinatorica 15 (1995), no. 1, 123–134] and Kaiser [Discrete Comput. Geom. 18 (1997), no. 2, 195–203].

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Acknowledgment

This material is based upon work supported by the National Science Foundation under Grant No. DMS-1440140 while the authors were in residence at the Mathematical Sciences Research Institute in Berkeley, California, during the Fall 2017 semester.

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Correspondence to Shira Zerbib.

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Frick, F., Zerbib, S. Colorful Coverings of Polytopes and Piercing Numbers of Colorful d-Intervals. Combinatorica 39, 627–637 (2019). https://doi.org/10.1007/s00493-018-3891-1

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Mathematics Subject Classification (2010)

  • 55M20
  • 52B11
  • 05B40
  • 52A35