Research notes on every problem, and a library of the papers behind them. Built from the open erdos repository.
Updated
Problem 586
claims/: The 1 claim page of Problem 586, one per claimant's result; the problem's standing derives from them.
Statement. Is there a covering system such that no two of the moduli divide each other?
Status. Disproved. The status-defining source is Theorem 1.2 of Balister, Bollobás, Morris, Sahasrabudhe and Tiba (Invent. Math. 228 (2022), 377--414, refereed): every finite covering system with moduli above has two moduli with one dividing the other, so the answer is no; the claim page is Balister, Bollobás, Morris, Sahasrabudhe and Tiba (accepted on the refereed publication and the site's credit).
Source. erdosproblems.com/586, accessed 2026-09-04 and re-read 2026-10-07 (no comments; empty proof-claim tab). Cite as: T. F. Bloom, Erdős Problem #586, https://www.erdosproblems.com/586.
References.
- [BBMST22] Balister, Paul and Bollobás, Béla and Morris, Robert and Sahasrabudhe, Julian and Tiba, Marius, On the Erdős covering problem: the density of the uncovered set. Invent. Math. 228 (2022), no. 1, 377-414, doi:10.1007/s00222-021-01087-5; arXiv:1811.03547 (2018). Held as balister_2018_erdos_covering_problem_density_uncovered_set.
Formalization. Statement in
formal-conjectures
(added 2026-09-20 and pinned to that commit), whose entry carries the category
research solved and a formal_proof attribute pointing to
src/latest/ErdosProblems/Erdos586.lean of Boris Alexeev's lean-proofs
repository at a pinned commit (read; the site's page as accessed recorded no formalization). That file declares
itself a formalization of the authors' theorem and is pinned on
the claim page;
nothing was built or audited here, and no local kernel credit is claimed.
Progress
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Linked library material
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