The points through are all at the same distance from the origin (call it ), as are the points through (call it ). We know probabilistically that and , so in particular . This observation “rules out” all but four rows of the table as possible colorings. (Pardon the imprecise nature of my statement; I’m quite tired.) These are the first two rows of the “top half” of the table, as well as the first two rows of the “bottom half”.

However, if we then look at points through (at a common distance from the origin), we must similarly have that , which rules out those four rows.

In other words, a computer analysis of this 420-vertex graph can produce a contradiction via the probabilistic approach. This is not remarkable necessarily, because we already have a smaller graph analyzed in a different way that produces a contradiction (the 278-vertex and the coloring of its central ), but it may be possible to minimize this automatically to get a smaller graph.

But I tried to do so and failed, similar to my other minimization attempts mentioned elsewhere in these comments.

]]>Specifically, I tried taking a 5-chromatic unit distance graph G that has, say, a copy of H at the origin. We know that the SAT problem of “properly 4-color G” is unsatisfiable, so the problem “properly 4-color G, and also have the central copy of H colored as 2tri” is also unsatisfiable. We can then minimize the graph using SAT minimization techniques.

The graphs I found in this way were larger and more cumbersome than Marijn’s graphs without the extra assumption. I was trying this before I learned about the many clever techniques Marijn describes in his paper, so it would be worthwhile to try again with those in mind. I also mentioned the idea to Marijn at some point.

]]>The 2.4 x 2.4 square is the largest square the SAT solver succeeded in 6-coloring, of any size pixel.

Re Warren at https://dustingmixon.wordpress.com/2018/06/16/polymath16-seventh-thread-upper-bounds/#comment-4839 : There was no successful 6-coloring on a torus (of course — because it would have been loudly announced, and also it might contradict some theorems), nor even on a square that is periodic in a single direction. (For reference, it was easy to 7-color a torus with even moderately small tiles, like side 1/6.)

Re Aubrey at https://dustingmixon.wordpress.com/2018/06/16/polymath16-seventh-thread-upper-bounds/#comment-4830 : I tried exactly the iterative approach you describe. I don’t know that I would say it was either better or worse than trying non-iteratively. Trying your “seed” would indeed be interesting.

Re Aubrey at https://dustingmixon.wordpress.com/2018/06/16/polymath16-seventh-thread-upper-bounds/#comment-4832 : I did not get a single UNSAT on any 6-coloring problem except for unreasonably large tiles (like 1/6 or 1/5 sidelength).

Re Aubrey at https://dustingmixon.wordpress.com/2018/06/16/polymath16-seventh-thread-upper-bounds/#comment-4833 : Yes, that idea occurred to me today, and I’d be excited to try it.

Re Warren at https://dustingmixon.wordpress.com/2018/06/16/polymath16-seventh-thread-upper-bounds/#comment-4843 : I agree that hex pixels would be a good idea. Whether the outer boundary is a circle or a square doesn’t concern me much, as I was trying my searches before Dustin announced a circle as the shape of choice. Also, as mentioned above, my attempts to use brute force to prove UNSAT for large areas were not successful.

]]>graph (point set) might NOT have a periodic coloring (or coloring might

have a different period). One can still make interesting statements, but must

keep this error in mind. ]]>

bounds. ]]>

really finite since the geometric configuration is periodic in the x-direction

(or perhaps “flip periodic” i.e. embed on a Mobius strip instead of a cylinder)!!

Why? Because this kind of graph can have more edges with same number of vertices,

that is why, causing a tendency to have greater chromatic numbers without

greater graph vertex count. Equivalently same chromatic numbers with smaller vertex count. Attention M.Heule!

to reduce your set in natural ways, only search for “nice” sets (with nice

polynomial descriptions) etc, yielding smaller searches snd more human-compatible results of them. And probably you can add more entries to the “dictionary” too. Definitely deserves more work, I clearly have inly scratched surface. ]]>

so what? Will also be able to say instead of “radius greater than 2”,

“width greater than 1.5” (or whatever value it is)

Now to make 1 further comment: if when doing the outer binary search for the max radius, or width, or whatever, you HALVE the pixel size every step of

the binary, then your total runtime will be pretty much the same as the single

last step in the binary search, with essentially same result as doing

the whole search.