Erdős Problem #348 — open problem in number theory

Erdős Problem #348

For what values of 0 ≤ m < n is there a complete sequence A = a_1 ≤ a_2 ≤ ⋯ of integers such that 1. A remains complete after removing any m elements, but 2. A is not complete after removing any n elements.

From the catalogue. Imported from The Formal Conjectures Authors (Google DeepMind and contributors) (Apache-2.0) — original. Nobody has started on it here yet: tasks are created as soon as someone asks for one or submits a claim. A Lean proof is checked against the statement below by the Lean kernel; a curator confirms before the problem counts as resolved.

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Cite

@misc{cairn-erdos-348,
  title        = {Erdős Problem #348},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/erdos-348}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-09-28}
}

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Claims

Verified

Disputed

Refuted

On the literature board

The problem

The question

For what values of $0 \leq m < n$ is there a complete sequence $A = {a_{1} \leq a_{2} \leq \dots}$ of integers such that

$A$ remains complete after removing any $m$ elements, but
$A$ is not complete after removing any $n$ elements.

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.ErdosProblems.«348». answer(sorry) marks a yes/no question: a proof of the statement on the right of ↔, or of its negation, answers it.

theorem erdos_348 :
    { (m, n) | (m) (n) (_ : m < n) (a : ℕ → ℕ) (_ : Monotone a)
      (_ : ∀ s, s.card = m → IsAddCompleteNatSeq' (Function.updateFinset a s 0))
        (_ : ∀ t, t.card = n → ¬IsAddCompleteNatSeq' (Function.updateFinset a t 0)) } =
    answer(sorry)

What counts as progress

A Lean proof of the pinned statement (or of its negation, for a yes/no question) — checked by the Lean kernel against the upstream statement; a curator confirms before the problem is marked resolved.
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/348. Report it as a literature claim.
A precise flaw in the formal statement (a misformalisation) — report it upstream too.

References

erdosproblems.com/348

Source and licence

Imported from Formal Conjectures (Erdős problems), commit e6d1743831c2. Statements and descriptions © The Formal Conjectures Authors, Apache License 2.0; reformatted for this page.