Document Open Access Logo

Three-Chromatic Geometric Hypergraphs

Authors Gábor Damásdi , Dömötör Pálvölgyi

Thumbnail PDF


  • Filesize: 0.72 MB
  • 13 pages

Document Identifiers

Author Details

Gábor Damásdi
  • MTA-ELTE Lendület Combinatorial Geometry Research Group, Dept. of Computer Science, ELTE Eötvös Loránd University, Budapest, Hungary
Dömötör Pálvölgyi
  • MTA-ELTE Lendület Combinatorial Geometry Research Group, Dept. of Computer Science, ELTE Eötvös Loránd University, Budapest, Hungary

Cite AsGet BibTex

Gábor Damásdi and Dömötör Pálvölgyi. Three-Chromatic Geometric Hypergraphs. In 38th International Symposium on Computational Geometry (SoCG 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 224, pp. 32:1-32:13, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)


We prove that for any planar convex body C there is a positive integer m with the property that any finite point set P in the plane can be three-colored such that there is no translate of C containing at least m points of P, all of the same color. As a part of the proof, we show a strengthening of the Erdős-Sands-Sauer-Woodrow conjecture. Surprisingly, the proof also relies on the two dimensional case of the Illumination conjecture.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Hypergraphs
  • Discrete geometry
  • Geometric hypergraph coloring
  • Decomposition of multiple coverings


  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    PDF Downloads


  1. Eyal Ackerman, Balázs Keszegh, and Dömötör Pálvölgyi. Coloring hypergraphs defined by stabbed pseudo-disks and ABAB-free hypergraphs. SIAM J. Discrete Math., 34(4):2250-2269, 2020. URL:
  2. Eyal Ackerman, Balázs Keszegh, and Máté Vizer. Coloring points with respect to squares. Discrete Comput. Geom., 58(4):757-784, 2017. URL:
  3. Greg Aloupis, Jean Cardinal, Sébastien Collette, Stefan Langerman, David Orden, and Pedro Ramos. Decomposition of multiple coverings into more parts. Discrete Comput. Geom., 44(3):706-723, 2010. URL:
  4. Andrei Asinowski, Jean Cardinal, Nathann Cohen, Sébastien Collette, Thomas Hackl, Michael Hoffmann, Kolja Knauer, Stefan Langerman, MichałLasoń, Piotr Micek, Günter Rote, and Torsten Ueckerdt. Coloring hypergraphs induced by dynamic point sets and bottomless rectangles. In Algorithms and data structures, volume 8037 of Lecture Notes in Comput. Sci., pages 73-84. Springer, Heidelberg, 2013. URL:
  5. Károly Bezdek and Muhammad A. Khan. The geometry of homothetic covering and illumination. In Discrete geometry and symmetry, volume 234 of Springer Proc. Math. Stat., pages 1-30. Springer, Cham, 2018. URL:
  6. Nicolas Bousquet, William Lochet, and Stéphan Thomassé. A proof of the Erdős-Sands-Sauer-Woodrow conjecture. J. Combin. Theory Ser. B, 137:316-319, 2019. URL:
  7. Jean Cardinal, Kolja Knauer, Piotr Micek, and Torsten Ueckerdt. Making triangles colorful. J. Comput. Geom., 4(1):240-246, 2013. URL:
  8. Jean Cardinal, Kolja Knauer, Piotr Micek, and Torsten Ueckerdt. Making octants colorful and related covering decomposition problems. SIAM J. Discrete Math., 28(4):1948-1959, 2014. URL:
  9. Jean Cardinal, Piotr Micek Kolja Knauer, Dömötör Pálvölgyi, Torsten Ueckerdt, and Narmada Varadarajan. Colouring bottomless rectangles and arborescences. To appear, 2020. Google Scholar
  10. Xiaomin Chen, János Pach, Mario Szegedy, and Gábor Tardos. Delaunay graphs of point sets in the plane with respect to axis-parallel rectangles. Random Structures Algorithms, 34(1):11-23, 2009. URL:
  11. Gábor Damásdi and Dömötör Pálvölgyi. Unit disks hypergraphs are three-colorable. Extended Abstracts EuroComb 2021, Trends in Mathematics, 14:483-489, 2021. Google Scholar
  12. Gábor Damásdi and Dömötör Pálvölgyi. Realizing an m-uniform four-chromatic hypergraph with disks. Combinatorica, to appear, 2022. Google Scholar
  13. Matt Gibson and Kasturi Varadarajan. Optimally decomposing coverings with translates of a convex polygon. Discrete Comput. Geom., 46(2):313-333, 2011. URL:
  14. Balázs Keszegh. Coloring half-planes and bottomless rectangles. Comput. Geom., 45(9):495-507, 2012. URL:
  15. Balázs Keszegh and Dömötör Pálvölgyi. Octants are cover-decomposable. Discrete Comput. Geom., 47(3):598-609, 2012. URL:
  16. Balázs Keszegh and Dömötör Pálvölgyi. Convex polygons are self-coverable. Discrete Comput. Geom., 51(4):885-895, 2014. URL:
  17. Balázs Keszegh and Dömötör Pálvölgyi. An abstract approach to polychromatic coloring: shallow hitting sets in ABA-free hypergraphs and pseudohalfplanes. J. Comput. Geom., 10(1):1-26, 2019. URL:
  18. Balázs Keszegh and Dömötör Pálvölgyi. Proper coloring of geometric hypergraphs. Discrete Comput. Geom., 62(3):674-689, 2019. URL:
  19. István Kovács. Indecomposable coverings with homothetic polygons. Discrete Comput. Geom., 53(4):817-824, 2015. URL:
  20. F. W. Levi. Überdeckung eines Eibereiches durch Parallelverschiebung seines offenen Kerns. Arch. Math. (Basel), 6:369-370, 1955. URL:
  21. János Pach. Decomposition of multiple packing and covering. Diskrete Geometrie, 2. Kolloq. Math. Inst. Univ. Salzburg, pages 169-178, 1980. Google Scholar
  22. János Pach. Covering the plane with convex polygons. Discrete Comput. Geom., 1(1):73-81, 1986. URL:
  23. János Pach and Dömötör Pálvölgyi. Unsplittable coverings in the plane. Adv. Math., 302:433-457, 2016. URL:
  24. János Pach, Dömötör Pálvölgyi, and Géza Tóth. Survey on decomposition of multiple coverings. In Geometry - intuitive, discrete, and convex, volume 24 of Bolyai Soc. Math. Stud., pages 219-257. János Bolyai Math. Soc., Budapest, 2013. URL:
  25. János Pach and Gábor Tardos. Coloring axis-parallel rectangles. J. Combin. Theory Ser. A, 117(6):776-782, 2010. URL:
  26. János Pach, Gábor Tardos, and Géza Tóth. Indecomposable coverings. In Discrete geometry, combinatorics and graph theory, volume 4381 of Lecture Notes in Comput. Sci., pages 135-148. Springer, Berlin, 2007. URL:
  27. Dömötör Pálvölgyi. Indecomposable coverings with concave polygons. Discrete Comput. Geom., 44(3):577-588, 2010. URL:
  28. Dömötör Pálvölgyi and Géza Tóth. Convex polygons are cover-decomposable. Discrete Comput. Geom., 43(3):483-496, 2010. URL:
  29. Ioannis Papadoperakis. An estimate for the problem of illumination of the boundary of a convex body in E³. Geom. Dedicata, 75(3):275-285, 1999. URL:
  30. A Prymak. Every 3-dimensional convex body can be covered by 14 smaller homothetic copies. arXiv preprint, 2021. URL:
  31. Bill Sands, Norbert W. Sauer, and Robert E. Woodrow. On monochromatic paths in edge-coloured digraphs. J. Combin. Theory Ser. B, 33(3):271-275, 1982. URL:
  32. Shakhar Smorodinsky and Yelena Yuditsky. Polychromatic coloring for half-planes. J. Combin. Theory Ser. A, 119(1):146-154, 2012. URL:
  33. Gábor Tardos and Géza Tóth. Multiple coverings of the plane with triangles. Discrete Comput. Geom., 38(2):443-450, 2007. URL:
Questions / Remarks / Feedback

Feedback for Dagstuhl Publishing

Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail