---

Claim. In P. Erdős and I. Z. Ruzsa, On the small sieve. I. Sifting by primes, J. Number Theory 12 (1980), no. 3, 385--394, $F(x,A)$ counts the $n\le x$ divisible by no element of $A$, $G(x,K)$ is the least $F(x,P)$ over sets $P$ of primes with $\sum1/p\le K$, and $H_m(x,K)$ is the least $F(x,A)$ over sets $A$ not containing $1$ with $\sum1/a\le K$, any $m$ of whose elements are coprime (displays (1.5) and (1.9), pp. 386--387). Theorem 3 proves $H_m(x,K)\ge c(m,K)x$ with $c(m,K)>0$ (display (1.10), printed with $\le$, a misprint: Section 4 derives it from the lower bound of Lemma 4.1). The authors note that the proof gives $H_m(x,K)\ge c_2e^{-K}G(x,K)$ for $x>x_0(m,K)$ (1.11). They then assert, "with a slight modification we can even prove" (p. 387), that $H_m(x,K)\ge G(x,K)-\epsilon x$ for $x>x_0(\epsilon,m,K)$ (1.12). The modification is not written out. For $m=2$ these are this problem's pairwise coprime sets, and sets of primes are admissible, so (1.12) would make the problem's minimum $G(N,C)+o(N)$: the corrected Statement reduces to the prime case.

Covers. The reduction of the corrected Statement to the case in which $A$ consists of primes. The prime case, $G(x,K)=(\rho(e^K)+o(1))x$, is the paper's Problem 1; it was unproved in 1980 and was later proved by Hildebrand 1987, so (1.12) and that corollary together would give the corrected Statement, which Tao 2026 proves with a written argument. Not covered: the exact minimizer for a given $N$, the site's wording. Theorem 3 alone gives only a positive proportion and settles no instance of the problem.

Standing. Claimed. The paper is refereed, but (1.12) is asserted without proof, so the refereeing does not vouch for it. No reviewer is recorded as having checked the modification, and the site's commentary and Tao's note cite the paper only for the prime-case question.

Date. The paper appeared in the August 1980 issue, volume 12, no. 3; the page name uses the first of that month.

Depends on. No page of this wiki.