Progress on the Hadwiger-Nelson problem! Thanks to a computer-assisted search, a finite collection of points in the plane has been located such that it is not possible to 4-color the points so that pairs of points a unit distance apart always have distinct colors. Hence the possible values of the chromatic number of the plane has now been narrowed down to 5, 6, or 7. (EDIT: the graph was found by hand, but the verification of non-4-colorability was computer assisted.)

The author (Aubrey de Grey, who incidentally was an amateur mathematician that had contributed to some previous Polymath projects) is interested in starting a project to reduce the amount of computer assistance required (right now one has to check that certain graphs with a couple thousand vertices are not 4-colorable, which is beyond the capability of human verification). For those interested in this, he can be reached on twitter at https://twitter.com/aubreydegrey

UPDATE: There is now a polymath proposal on this at polymathprojects.org/2018/04/10/polymath-proposal-finding-simpler-unit-distance-graphs-of-chromatic-number-5/
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