We have not worked much on spindle density, but one interesting feature is that the spindle density I was able to achieve is actually a bit lower for the 5-chromatic graphs arising from the 30 unit vectors in my graph V than for the graphs arising from the original 18 unit vectors (those in the graph U). I’m unaware of any attempt to identify a theoretical upper bound.

Also, note that the upper bound on e is actually o(1) more than the 4/3 power of v. I believe that only one graph has ever been discovered that exceeds 4/3, though – it has 16 vertices and 41 edges and is due to Ed Pegg:

https://math.stackexchange.com/questions/2575268/maximally-dense-unit-distance-graphs

]]>Maybe you know about new fractional CN record, 3.8992: https://arxiv.org/abs/1810.00960.

Thanks to Aubrey and D.E. Knuth I finally understand how to calculate CN for large graphs using SAT, but I still don’t know how to get fractional CN. Probably it will be interesting to calculate it for known 5- or almost-5-chromatic graphs.

I reread first threads trying to find the answer, what does it take to be 5-chromatic? There is a lot of thoughts here, which I cannot understand before, but I still haven’t a clear picture. What is essential here? Spindles density, edges density, vertices number, rotations set, something else?

I plan to define spindles density for known graphs. Is there some theoretic bound for spindles density ? How does it relate to edges density ? Here are the numbers of Moser’s spindles, edges and vertices. The best known bound for edges is , see interesting pictures here:

https://math.stackexchange.com/questions/1986369/erdos-unit-distance-problem-maximal-graphs?rq=1 ]]>

I’m still working on my plan to make some actual plots of how certain variables affect chromatic number, based on the really helpful posts from Jaan and Tom in the previous thread. I feel though that this still might take the one or the other month to really bear any demonstrable fruits.

Aside from that, now that the n-ball graph approach to explaining why tile-based colorings are necessary feels rather complete, I’ve returned to tackling the topological approach to the same issue, using punctured neighborhoods again.

It has felt too deceptively simple the last time I seriously worked on it (at the start of this year), but putting the n-ball approach together has given me new certainty that there’s mileage to that approach too.

Things have gone a bit quiet here, so I wonder if it might be useful if everyone could post a little update on the status of their work. I’m hoping that quite a few of you are quietly beavering way on one or another aspect of the approaches we’ve been exploring. Don’t be shy!

]]>Let the tile coloured black in the left-hand panel be A and the one coloured black in the right-hand panel be B. Then place two A’s symmetrically at the North pole of the sphere, so that the pole sits at the centre of their shared edge. Do the same at the South pole and at four points on the equator, and orient them perpendicular to each other in the same way as for the plane. Then each face of the octahedron has three of tile B meeting in its centre and three halves of tile A at its corners.

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