This is the tenth “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.
Here’s a brief summary of the progress made in the previous thread:
– We have new results in the probabilistic formulation, namely, Proposition 36 and Lemmas 38 and 39.
– Jaan Parts refined Pritikin’s analysis to prove that every unit-distance graph with at most 24 vertices is 5-colorable, and every such graph with at most 6906 vertices is 6-colorable.
– Domotor, Frankl and Hubai showed that, if there exists a k-chromatic unit-distance graph with a bichromatic origin such that all its neighbors are in the upper half-plane and all their coordinates are rational, then CNP is at least k.
At the moment, there are a few outstanding SAT instances that we’d like to resolve:
– Can we find a tile-based 6-coloring of the plane? Such a coloring must satisfy several necessary conditions.
– Aubrey has suggested a new take on Boris’ pixelated colorings by SAT solver. I have heard offline that the details of this approach are evolving.