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].
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
R. Aharoni, T. Kaiser and S. Zerbib: Fractional covers and matchings in families of weighted d-intervals, Combinatorica 37 (2017), 555–572.
M. Asada, F. Frick, V. Pisharody, M. Polevy, D. Stoner, L. Tsang and Z. Wellner: Fair division and generalizations of Sperner- and KKM-type results, SIAM J. Discrete Math. 32 (2018), 591–610.
E. Babson: Meunier conjecture, arXiv preprint arXiv:1209.0102 (2012).
R. B. Bapat: A constructive proof of a permutation-based generalization of Sperner’s lemma, Math. Program. 44 (1989), 113–120.
I. Bárány: A generalization of Carathéodory’s theorem, Discrete Math. 40 (1982), 141–152.
J. A. De Loera, E. Peterson and F. E. Su: A polytopal generalization of Sperner’s lemma, J. Combin. Theory, Ser. A 100 (2002), 1–26.
F. Frick, K. Houston-Edwards and F. Meunier: Achieving rental harmony with a secretive roommate, Amer. Math. Monthly, to appear.
Z. Füredi: Maximum degree and fractional matchings in uniform hypergraphs, Combinatorica 1 (1981), no. 2, 155–162.
D. Gale: Equilibrium in a discrete exchange economy with money, Int. J. Game Theory 13 (1984), no. 1, 61–64.
T. Kaiser: Transversals of d-intervals, Discrete Comput. Geom. 18 (1997), no. 2, 195–203.
B. Knaster, C. Kuratowski and S. Mazurkiewicz: Ein Beweis des Fixpunktsatzes für n-dimensionale Simplexe, Fund. Math. 14 (1929), 132–137.
H. Komiya: A simple proof of KKMS theorem, Econ. Theory 4 (1994), 463–466.
J. Matoušek: Lower bounds on the transversal numbers of d-intervals, Discrete Comput. Geom. 26 (2001), 283–287.
F. Meunier, W. Mulzer, P. Sarrabezolles and Y. Stein: The rainbow at the end of the line: a PPAD formulation of the colorful Carathéodory theorem with applications, Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms, Society for Industrial and Applied Mathematics, 2017, 1342–1351.
O. R. Musin: KKM type theorems with boundary conditions, J. Fixed Point Theory Appl. 19 (2017), 2037–2049.
L. S. Shapley: On balanced games without side payments, Math. Program., Math. Res. Center Publ. (T. C. Hu and S. M. Robinson, eds.), vol. 30, Academic Press, New York, 1973, 261–290.
M. Shih and S. Lee: Combinatorial formulae for multiple set-valued labellings, Math. Ann. 296 (1993), no. 1, 35–61.
F. E. Su: Rental harmony: Sperner’s lemma in fair division, Amer. Math. Monthly 106 (1999), 930–942.
G. Tardos: Transversals of 2-intervals, a topological approach, Combinatorica 15 (1995), 123–134.
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.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
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
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00493-018-3891-1