A solution to Ringel's circle problem

TL;DR

This paper constructs circle families with complex tangency graphs to answer Ringel's circle problem negatively, using advanced combinatorial theorems and polynomial constraints.

Contribution

It introduces a multidimensional Gallai's theorem with polynomial constraints derived from the Hales-Jewett theorem, providing a novel approach to circle graph problems.

Findings

Constructed circle families with arbitrarily large girth and chromatic number.

Provided a strong negative answer to Ringel's circle problem.

Developed a new multidimensional Gallai's theorem with polynomial constraints.

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.