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A counterexample to the Ringel circle problem: arxiv.org/abs/2112.05042, via gilkalai.wordpress.com/2021/12

Ringel asked whether systems of circles, tangent only in pairs, can be colored with O(1) colors so no two tangent circles have the same color. Five colors were known to be necessary but unknown to be sufficient; see the lead image of my old web page ics.uci.edu/~eppstein/junkyard

Now Davies, Keller, Kleist, Smorodinsky, and Walczak have found systems of circles requiring arbitrarily many colors.

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