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A Solution to Ringel’s Circle Problem

Authors James Davies, Chaya Keller , Linda Kleist , Shakhar Smorodinsky , Bartosz Walczak

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Author Details

James Davies
  • University of Waterloo, Canada
Chaya Keller
  • Ariel University, Israel
Linda Kleist
  • Technische Universität Braunschweig, Germany
Shakhar Smorodinsky
  • Ben-Gurion University of the Negev, Beer-Sheva, Israel
Bartosz Walczak
  • Department of Theoretical Computer Science, Faculty of Mathematics and Computer Science, Jagiellonian University, Kraków, Poland


This work was initiated at the online workshop "Geometric graphs and hypergraphs". We thank the organizers Torsten Ueckerdt and Yelena Yuditsky for a very nice workshop and all participants for fun coffee breaks and a fruitful atmosphere.

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James Davies, Chaya Keller, Linda Kleist, Shakhar Smorodinsky, and Bartosz Walczak. A Solution to Ringel’s Circle Problem. In 38th International Symposium on Computational Geometry (SoCG 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 224, pp. 33:1-33:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)


We construct families of circles in the plane such that their tangency graphs have arbitrarily large girth and chromatic number. This provides a strong negative answer to Ringel’s circle problem (1959). The proof relies on a (multidimensional) version of Gallai’s theorem with polynomial constraints, which we derive from the Hales-Jewett theorem and which may be of independent interest.

Subject Classification

ACM Subject Classification
  • Theory of computation → Computational geometry
  • circle arrangement
  • chromatic number
  • Gallai’s theorem
  • polynomial method


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