Problem 27

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claims/: The 2 claim pages of Problem 27, one per claimant's result; the problem's standing derives from them.

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Statement. An $\epsilon$-almost covering system is a set of congruences $a_i\pmod{n_i}$ for distinct moduli $n_1<\cdots<n_k$ such that the density of those integers which satisfy none of them is $\leq \epsilon$.

Is there a constant $C>1$ such that for every $\epsilon>0$ and $N\geq 1$ there is an $\epsilon$-almost covering system with $N\leq n_1<\cdots <n_k\leq CN$?

Formulation. The question as worded asks for one $C$ that works for every pair $(\epsilon,N)$, so a given $C$ fails as soon as one pair admits no $\epsilon$-almost covering system. Hough's minimum-modulus theorem ([Ho15]; bound $616000$ by [BBMST22]) alone supplies such a pair: for $N$ beyond its bound, the finitely many systems with distinct moduli in $[N,CN]$ all fail to cover, and each leaves a periodic uncovered set of density at least one over the least common multiple of its moduli, so every smaller $\epsilon$ is too small. The conjecture Erdős and Graham made, and the one Erdős offered a prize for, is the uniform form: for each $C>1$ a positive $d_C$ such that, for all large $N$, every choice of residue classes with distinct moduli in $[N,CN]$ leaves density at least $d_C$ uncovered. That form is what [FFKPY07] prove (their Theorem B, with any $d_C<1/C$ admissible) and what Theorem 5.1 of [BBMST22] proves again with a threshold on $N$ independent of $C$; both claim pages record the uniform form.

Status. Disproved, the site's label. The answer is no, by Filaseta, Ford, Konyagin, Pomerance and Yu (J. Amer. Math. Soc. 2007, refereed), whom the site credits; the acceptance evidence is on their claim page. A second proof, Theorem 5.1 of Balister, Bollobás, Morris, Sahasrabudhe and Tiba (Invent. Math. 2022, refereed), has its own claim page.

Source. erdosproblems.com/27, accessed 2026-09-04 and 2026-10-07 (page last edited 16 July 2026; two editorial comments of 12 and 13 July 2026 on the wording of the commentary, and no proof claim, on its thread). Cite as: T. F. Bloom, Erdős Problem #27, https://www.erdosproblems.com/27.

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. (2022), 377-414.
• [FFKPY07] Filaseta, Michael and Ford, Kevin and Konyagin, Sergei and Pomerance, Carl and Yu, Gang, Sieving by large integers and covering systems of congruences. J. Amer. Math. Soc. (2007), 495-517.
• [Ho15] Hough, Bob, Solution of the minimum modulus problem for covering systems. Ann. of Math. (2) (2015), 361-382.

Formalization. The site's indicator reads "Formalised statement? No" and the community database records the problem as not formalized (2026-10-07). The locatable artifact is the Lean development src/latest/ErdosProblems/Erdos27.lean of Boris Alexeev's lean-proofs repository (added 2026-08-17; pinned on the Filaseta–Ford–Konyagin–Pomerance–Yu claim page as a formalization of their solution), which states the site's question under its own definitions and proves its negation. This corpus has not built or checked it, and no local kernel credit is claimed.

Progress

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Known Results

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Linked library material

These entries are derived from explicit links on library pages. They are navigation only and do not by themselves record mathematical progress.

• balister_2018_erdos_covering_problem_density_uncovered_set
• balister_2018_erdos_covering_problem_density_uncovered_set / theorem_5_1
• filaseta_2007_sieving_large_integers_covering_systems_congruences
• filaseta_2007_sieving_large_integers_covering_systems_congruences / theorem_4
• filaseta_2007_sieving_large_integers_covering_systems_congruences / theorem_b