---

Claim. Let $A$ be an infinite set of pairwise coprime positive integers with property P. Schoen's Theorem (p. 193) gives $|A\cap\{1,\ldots,N\}|<2N^{2/3}$ for infinitely many $N$, 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. Schoen'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. The squares of the primes $p\equiv3\pmod4$ show that the exponent cannot go below $1/2$ (p. 195).

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: J. Combin. Theory Ser. A 94 (2001), no. 1, 191--195.

Depends on. No page of this wiki.