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Claim. Theorem 1 of R. Ahlswede and L. H. Khachatrian, Maximal sets of numbers not containing $k+1$ pairwise coprime integers, Acta Arith. 72 (1995), no. 1, 77--100, carded at Ahlswede and Khachatrian 1995: with $f(N,k)$ the largest size of a set $A\subseteq\{1,\ldots,N\}$ with no $k+1$ pairwise coprime elements and $E(N,k)$ the set of integers up to $N$ divisible by one of the first $k$ primes, for every $k$ there is $n(k)$ such that $f(N,k)=|E(N,k)|$ for every $N>n(k)$, and $E(N,k)$ is the only set of that size. The authors call this the strongest statement one can hope for after the counterexample of their 1994 paper. The paper also states two weaker density forms: Theorem 1A, that the supremum of the lower densities of sets of positive integers with no $k+1$ pairwise coprime elements is the density $1-\prod_{i\le k}(1-1/p_i)$ of the multiples of the first $k$ primes, which now follows from Theorem 1 and is given its simpler original proof; and Theorem 1B, that $f(N,k)/|E(N,k)|\to1$ as $N\to\infty$ for every $k$, whose original proof the paper omits. The proof of Theorem 1 rests on Theorem 2, a shadow inequality for families of $l$-sets with no $k+1$ pairwise disjoint members.

Covers. The instances $(k,N)$ of Problem 56 with $N>n(k)$, each answered yes. The stated question, which asks this for every $N\geq p_k$, stays answered no by the 1994 claim. Erdős's stronger form of the follow-up, recorded on the site from [Er92b] and [Er95], asks whether the conjecture holds once $N\geq(1+o(1))p_k^2$; the theorem gives no such bound on $n(k)$, so that form is not settled.

Depends on. Nothing in this wiki; the result is the paper's own theorem.

Acceptance. Refereed: Acta Arith. 72 (1995), no. 1, 77--100, a refereed journal. The site's DISPROVED (LEAN) label credits the 1994 paper, not this one, so no reviewed evidence is listed. The page is dated by the publication year alone: the publisher's record gives 1995 without a month, so the first day of the year stands in for the issue date.