This is the seventh “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:
– The smallest known unit-distance graph now has 553 vertices and 2722 edges.
– Terry Tao made some progress on understanding the colorings of .
There are several interesting ideas captured in Marijn’s paper. For example, his Table 1 appears to be amenable to analysis under the probabilistic method.
There has also been some work to find 6-colorings of the plane. Along these lines, I wanted to reiterate a point made by Matthew Kahle:
Theorem (Thomassen). Consider any infinite connected plane graph such that each connected component of (or “face” of ) has diameter less than 1, is homeomorphic to a disk, and is bounded by a cycle in . Next, consider any proper coloring of such that each color class is the union of faces of (and part of their boundaries) such that the distance between any two of these faces is greater than 1. All such colorings require at least 7 colors.
That is, “nice” colorings will not improve the current upper bound. Regardless, as Philip suggested, it would be interesting to collect some records along these lines:
Question. What is the radius of the largest disk in that can be 6-colored?