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Claim. Every irreducible covering set $1<n_1<\cdots<n_k$ of Problem 1189 satisfies $n_k\le2^{k-1}$. The paper, R. J. Simpson, Regular coverings of the integers by arithmetic progressions, Acta Arith. 45 (1985), no. 2, 145–152, calls a covering of the integers by arithmetic progressions regular when no proper subcollection covers, and its Corollary 2 proves Znám's conjecture: a regular covering whose moduli have least common multiple $N=\prod p^{\alpha_p}$ has at least $F(N)+1$ progressions, where $F(N)=\sum\alpha_p(p-1)$. The bound on $n_k$ is a corollary of this theorem, not a statement of the paper. Choose residues for an irreducible covering set so that the classes cover; the resulting system is regular, since a covering proper subsystem would make a proper subset of the moduli a covering set. Hence $F(N)\le k-1$, and since $p-1\ge1$ and $p\ge2$ give $F(N)\ge\sum\alpha_p\ge\log_2N$, the largest modulus satisfies $n_k\le N\le2^{F(N)}\le2^{k-1}$. A comment of 2026-08-17 on the problem's discussion thread makes the same remark, that the bound follows from the paper's theorem rather than being stated there. The source card is Simpson 1985, with the paper's Theorem 1 on its own page.

Covers. The upper bound $n_k\le2^{k-1}$ on the largest modulus of an irreducible covering set of size $k$, which the site credits to Simpson [Si85]. It does not give the exact maximum of $n_k$, its minimum, the count of irreducible covering sets or the maximum reciprocal sum.

Depends on. No page of this wiki; the inequality is the paper's.

Acceptance. Refereed: Acta Arithmetica 45 (1985), no. 2, 145–152, the DOI linked above; the publisher's record gives the year without a month, so the page is named by the first day of 1985. The site's commentary credits the bound to Simpson, but the site labels the problem OPEN, so that remark is context and not acceptance.