This is the second “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.

What follows is a summary of the progress made thus far:

**Simplifying de Grey’s graph.** There have been three strides along these lines. One is to decrease the size of the graph by iteratively removing vertices that are not necessary to be 5-chromatic. This approach has led to our current record holder, which has 826 vertices and 4273 edges. Another approach has been to hunt for more symmetric 5-chromatic graphs, in the hopes that such graphs might be easier to analyze by hand (e.g., see this and that). Finally, others have been hunting for extremely small alternatives to de Grey’s L and M graphs that are amenable to by-hand analysis, in the hopes that these would eventually lead to a 5-chromatic graph (e.g., see this and that).

Continue reading Polymath16, second thread: What does it take to be 5-chromatic?