O’Donnell Earns His Doctorate

I agree with Ron Graham, Paul O’Donnell proved sensational results, showing that there are unit distance 4-chromatic graphs of girth 9, girth 12, and even of an arbitrarily high girth. These results did give some evidence that, perhaps, the search for a 5-chromatic unit-distance graph may celebrate its victory one day – this is the result O’Donnell was ultimately after but had not succeeded, no one had until 2018 (later about it).