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Claim. Every finite covering system with pairwise distinct moduli greater than one has a modulus divisible by $2$ or by $3$. This is Theorem 1 of R. D. Hough and P. P. Nielsen, Covering systems with restricted divisibility, Duke Math. J. 168 (2019), no. 17, 3261--3295. The proof works in $\mathbb Z/Q\mathbb Z$ with $Q$ the least common multiple of the moduli and gives a positive lower bound for the density of the uncovered set (quantitatively, for related quantities) by sieving in stages over good fibres modulo partial least common multiples, using Shearer-type and Lovász-type lower bounds. The statement is recorded on the library's source card; the proof is not compiled there. Balister, Bollobás, Morris, Sahasrabudhe and Tiba later gave a simpler proof as Theorem 7.1 of their Inventiones paper, compiled on its library page and recorded with their further restriction on their claim page.

Covers. The case of Problem 7 in which no modulus is divisible by $3$: no finite covering system with distinct odd moduli greater than one, all coprime to $3$, exists. A hypothetical distinct odd covering must have a modulus divisible by $3$. The unrestricted question stays open.

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

Acceptance. Refereed: Duke Mathematical Journal 168 (2019), no. 17, 3261--3295, doi:10.1215/00127094-2019-0058; the arXiv v1 of 2017-03-06 names this page. Not reviewed: the site's commentary credits the theorem, but the site labels the problem VERIFIABLE, an open label, so the credit settles no part of the problem on the site's account. Not formalized: no Lean proof of the theorem is recorded.