Erdős Problem #1189

The question

erdos_1189.parts.i. Call a set of distinct integers $1<n_1<\cdots <n_k$ a covering set if there is a choice of $a_i(\mathrm{mod}{n}_i)$ for $1\le i\le k$ such that every integer satisfies at least one of these congruences. A set is an irreducible covering set if no proper subset is a covering set.

How many irreducible covering sets of size $k$ are there?

This is the first part of the questions of Erdős [Er80, p. 95]. We read "how many" as asking for the asymptotics of $\log I(k)$, where $I(k)$ is the number of irreducible covering sets of size $k$; this is the scale on which the number of minimal covering systems is known [BBMST24].

erdos_1189.parts.ii. Call a set of distinct integers $1<n_1<\cdots <n_k$ a covering set if there is a choice of $a_i(\mathrm{mod}{n}_i)$ for $1\le i\le k$ such that every integer satisfies at least one of these congruences. A set is an irreducible covering set if no proper subset is a covering set.

What is the minimum and maximum that $n_k$ can be?

This is the second part. Here $n_k$ is the largest element of an irreducible covering set of size $k$. The answer is a pair (minimum, maximum), stated for each $k$ for which an irreducible covering set of size $k$ exists.

erdos_1189.parts.iii. Call a set of distinct integers $1<n_1<\cdots <n_k$ a covering set if there is a choice of $a_i(\mathrm{mod}{n}_i)$ for $1\le i\le k$ such that every integer satisfies at least one of these congruences. A set is an irreducible covering set if no proper subset is a covering set.

Determine or estimate $\max \sum \frac{1}{n_i}$, where the maximum ranges over all irreducible covering sets of size $k$.

This is the third part. We read "determine or estimate" as an estimate up to constant factors. In [Er80, p. 95] the maximum is taken over the irreducible covering sets $1<n_1<\cdots <n_k\le x$, as a function of $x$; we follow the problem page, which takes it over the irreducible covering sets of size $k$.

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.ErdosProblems.«1189» (3 statements).

theorem erdos_1189.parts.i :
    (fun k : ℕ ↦ Real.log (irreducibleCount k : ℝ)) ~[atTop] erdos_1189.parts.i.answer

theorem erdos_1189.parts.ii (k : ℕ)
    (hk : ∃ M : Finset ℕ, IsIrreducibleCoveringSet M ∧ M.card = k) :
    IsLeast (largestModulus k) (erdos_1189.parts.ii.answer k).1 ∧
      IsGreatest (largestModulus k) (erdos_1189.parts.ii.answer k).2

theorem erdos_1189.parts.iii :
    maxReciprocalSum =Θ[atTop] erdos_1189.parts.iii.answer

What counts as progress

• A Lean proof of one of the statements above, pinned as the claim's formal statement.
• Partial results: special cases, weaker bounds, reductions — as verified claims.
• Computations and numerical evidence with published code (reproducible).
• Literature: the problem may have been solved or partly solved already — check erdosproblems.com/1189. Report it as a literature claim.
• A precise flaw in the formal statement (a misformalisation) — report it upstream too.

References

• erdosproblems.com/1189
• [Er80] Erdős, P., A survey of problems in combinatorial number theory. Ann. Discrete Math. 6 (1980), 89-115. PDF
• [Si85] Simpson, R. J., Regular coverings of the integers by arithmetic progressions. Acta Arith. 45 (1985), 145-152.
• [Su07] Sun, Z.-W., On covering numbers. Combinatorial Number Theory, de Gruyter (2007), 443-453. arXiv:math/0601017
• [BBMST24] Balister, P., Bollobás, B., Morris, R., Sahasrabudhe, J. and Tiba, M., *The structure and number of Erdős covering systems*. J. Eur. Math. Soc. 26 (2024), 75-109. arXiv:1904.04806

See also erdosproblems.com/1188, which counts minimal covering systems with bounded moduli.

Source and licence

Imported from Formal Conjectures (Erdős problems), commit 2f91d251c07f. Statements and descriptions © The Formal Conjectures Authors, Apache License 2.0; reformatted for this page.