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Claim. Let $A$ be an infinite set of pairwise coprime positive integers with property P. Baier's Theorem (p. 2) gives, for every $\varepsilon>0$, $|A\cap\{1,\ldots,N\}|<(3+\varepsilon)N^{2/3}(\log N)^{-1}$ for infinitely many $N$, sharpening Schoen's bound $2N^{2/3}$ (Schoen's claim page). So for every $c<1/3$ there are infinitely many $N$ with $|A\cap\{1,\ldots,N\}|<N^{1-c}$: for pairwise coprime sets the second question of Problem 12 has the answer yes. Baier's P-sets let the two larger elements coincide; on infinite pairwise coprime sets the two readings agree, since $a\mid2b$ with $\gcd(a,b)=1$ forces $a\le2$, and such a set contains neither $1$, which divides every sum, nor $2$, which divides the sum of any two of its other, odd, elements. Baier's introduction (p. 1) recalls Schoen's counterexample, the squares of the primes $p\equiv3\pmod4$, which shows that the exponent cannot go below $1/2$.

Covers. The second question for pairwise coprime sets. Not the second question in general, which the DeepMind claim page answers no, and not the first or third.

Acceptance. Refereed: Integers 4 (2004), #A13.

Depends on. No page of this wiki.