A solution to Ringel's circle problem
Combinatorics
2023-09-07 v2 Discrete Mathematics
Abstract
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.
Cite
@article{arxiv.2112.05042,
title = {A solution to Ringel's circle problem},
author = {James Davies and Chaya Keller and Linda Kleist and Shakhar Smorodinsky and Bartosz Walczak},
journal= {arXiv preprint arXiv:2112.05042},
year = {2023}
}
Comments
new section on an "induced" version of Gallai's theorem with constraints, minor other improvements