I have just uploaded to the arxiv a revision of my paper (I gather it will go live within a few hours), incorporating this correction as well as making the construction clearer in section 6 (and adding a number of particularly apposite acknowledgements!). This therefore leads to a size of 1581 for the full 5-chromatic graph (which contains two copies of Sa). Update: after fixing my bug and re-running the code that seeks deletions from Sa, I now find that just two vertices can be deleted wtthout introducing a problematic (as defined in section 5.4) 4-coloring. These and related problems are discussed also in my survey article: Some old and new problems in combinatorial geometry I: Around Borsuk’s problem. Let me also mention the related Rosenfeld’s problem discussed in this post: Let G be the graph whose vertices are points in the plane and two vertices form an edge if their distance is an odd integer. Is the chromatic number of this graph finite? (From Wikipedea: A seven-coloring of the plane, and a four-chromatic unit distance graph in the plane (the Moser spindle), providing upper and lower bounds for the Hadwiger–Nelson problem.) Below, two figures from Aubret de Grey’s paper. (April 21) A great blog post (French) on Automath. (April 19) An excellent article on Quanta Magazine by Evelyn Lamb. Let me also mention two related old posts over Lipton and Regan’s blog ( one, two). A post over Shtetl Optimized describes the new development along with another important development on quantum computation. A blog post reporting on independent verification of some of the new results is over Dustin G. Here is an earlier Google+ post by Terry Tao and an earlier blogpost on the new result by Jordan Ellenberg proposing to use the polynomial method to tackle the upper bound. The project is devoted to the chromatic number of the plane ( Wikipage) following Aubrey de Grey’s example showing that the chromatic number of the plane is at least 5. (April 14) Dustin Mixon and Aubrey de Grey have launched Polymath16 over at Dustin’s blog. Noam Elkies independently proposed over MathOverflow a polymath project following Aubrey de Grey’s paper. Of course, this project may lead to independent verification of the result, and perhaps even insights for what is needed to replace ‘5’ with ‘6’. ![]() Updates: Aubrey de Grey has made now a polymath proposal over the polymath blog aimed at finding simpler constructions, namely constructions with a smaller number of vertices, or where the computerized part of the proof is simpler. The Moser spindle is a simple example of a unit-distance graph with chromatic number 4, and there is a simple coloring of the plane, found by Isbell, based on the hexagonal packing with 7 colors so that no color contains a pair of points of distance 1.Īubrey de Grey has constructed a unit distance graph with 1567 vertices which is 5-chromatic! Untill recently it was known that the answer can be 4,5,6 or 7. The problem was posed in 1950 by Edward Nelson, and related results already appeared in a paper by Hugo Hadwiger from 1945. The answer is referred to as the chromatic number of the plane. The Hadwiger–Nelson problem asks for the minimum number of colors required to color the plane such that no two points at distance one from each other have the same color. A major progress on an old standing beautiful problem. Aubrey de Grey proved that the chromatic number of the plane is at least 5.
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