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Claim. With $f(N)$ the quantity of Problem 202 and $L(N)=\exp(\sqrt{\log N\log\log N})$,

$$NL(N)^{-\sqrt2-o(1)}<f(N)<NL(N)^{-1/6+o(1)}.$$

The lower bound is a Chinese-remainder construction with ordered prime-power factors, refining Erdős and Szemerédi's; the upper bound is the Corollary to Theorem 1, which reduces arbitrary distinct moduli to the squarefree case, where Theorem 1 gives the coefficient $1/2$. E. S. Croot III, On non-intersecting arithmetic progressions, Acta Arith. 110 (2003), no. 3, 233–238, first posted as arXiv:math/0208236 on 2002-08-30 and cited as [Cr03b] on the problem page; the library's lower bound, Theorem 1 and Corollary pages compile the proofs.

Covers. The bounds above, which improved Erdős and Szemerédi's $N\exp(-(\log N)^{1/2+\epsilon})<f(N)<N(\log N)^{-c}$ on their claim page. Both sides were later sharpened: the upper coefficient by Chen and both by de la Bretèche, Ford and Vandehey. The claim does not determine $f(N)$.

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

Acceptance. Refereed: Acta Arithmetica 110 (2003), no. 3, 233–238, doi:10.4064/aa110-3-3. Not reviewed: the site labels the problem SOLVED (LEAN) and credits the answer to Ho's result, not to this paper.