A Solution to Ringel's Circle Problem

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

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review


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.

Original languageEnglish
Title of host publication38th International Symposium on Computational Geometry, SoCG 2022
EditorsXavier Goaoc, Michael Kerber
PublisherSchloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
ISBN (Electronic)9783959772273
StatePublished - 1 Jun 2022
Event38th International Symposium on Computational Geometry, SoCG 2022 - Berlin, Germany
Duration: 7 Jun 202210 Jun 2022

Publication series

NameLeibniz International Proceedings in Informatics, LIPIcs
ISSN (Print)1868-8969


Conference38th International Symposium on Computational Geometry, SoCG 2022


  • Gallai's theorem
  • chromatic number
  • circle arrangement
  • polynomial method

ASJC Scopus subject areas

  • Software


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