This is the seventeenth “research” thread of the Polymath16 project to make progress on the Hadwiger–Nelson problem, continuing this post. This project is a follow-up to Aubrey de Grey’s breakthrough result that the chromatic number of the plane is at least 5. Discussion of the project of a non-research nature should continue in the Polymath proposal page. We will summarize progress on the Polymath wiki page.
At this point, we are finalizing a draft of a paper by D.H.J. Polymath. Like Aubrey’s original paper, this will be submitted for publication in Geombinatorics.
This post concludes the Polymath16 project. Of course, we anticipate that folks will continue to make progress on this problem on their own. If you’d like to update the Polymath16 community about your progress, feel free to comment on this post.
I asked Aubrey to offer a few words to reflect on our project:
The Polymath 16 project recently celebrated 1000 days of age. It had two main initial goals: to find 5-chromatic unit-distance graphs in the plane with fewer vertices than the 1581-vertex example that I had published in early 2018, and to identify proofs of that required less computer assistance than my construction needed (ideally none at all). Many ancillary goals were also mentioned at the outset.
By all standards, the project has been an immense success. The record for the smallest graph was progressively improved, including several times by Marijn Heule, and the present record is 509, achieved by Jaan Parts. Jaan also has the distinction of cracking the other main challenge: a few months ago he unveiled a proof of that he described only as “human-verifiable,” but that is too modest, because his method is certainly usable to create a proof from scratch without any computational assistance in no more than a week or two (for someone sufficiently diligent and non-error-prone!). Jaan has, in fact, been the most prolific of the contributors to the project over the years, having made numerous other contributions, some on his own and some in Polymath-esque rapid-fire collaboration with others.
Perhaps the most satisfying aspect of the project, to me, is its quite remarkable success in attracting amateur mathematicians like myself (of whom Jaan is one). At least half a dozen of us have been significant contributors over these three years, working side by side with numerous professionals including Terry Tao himself.
Given the diversity of Hadwiger-Nelson-type problems that we have investigated, it was always implausible that all our results would be gathered into a single publication. This was especially the case in view of the proportion (more than half) of our advances that were achieved by a single individual with little or no collaboration at the blog, and which were thus more properly published under that person’s name alone. However, one clear exception emerged: the set of questions concerning bounds on the sizes of circular disks and infinite strips of different chromatic number. It has turned out that at least five of us made major contributions to one or more such question, so this was the natural focus of what has just become the first (and, we must now anticipate, only) paper arising from Polymath 16 whose sole listed author is the canonical, fictional D.H.J. Polymath. Like several of the other reports, and like my original proof of , it will appear in the journal Geombinatorics, whose editor-in-chief Alexander Soifer was, via his timeless 2009 book, responsible for inspiring me to work on the Hadwiger-Nelson problem in the first place.
The energy that Polymath 16 has evoked will not dissipate easily. We plan to leave this blog open for further discussions and reports – which will surely emerge as time passes, since many fascinating Hadwiger-Nelson-type questions remain wide open in spite of our efforts. Let us hope that the merry band who have progressed this far will continue to grow and to make yet further inroads into this truly captivating family of questions.