Erdős Problem #312 — open problem in combinatorics

Erdős Problem #312

Does there exist a constant c > 0 such that, for any K > 1, whenever A is a sufficiently large finite multiset of integers with Σ_n ∈ A 1/n > K there exists some S ⊆ A such that 1 - exp(-(c*K)) < Σ_n ∈ S 1/n ≤ 1?

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-312,
  title        = {Erdős Problem #312},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/erdos-312}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-09-29}
}

Also: CITATION.cff · Atom feed of results

Claims

Verified

Disputed

Refuted

On the literature board

The problem

The question

Does there exist a constant c > 0 such that, for any K > 1, whenever A is a sufficiently large finite multiset of integers with $\sum_{n \in A} 1/ n > K$ there exists some $S \subseteq A$ such that $1 - exp (- (c * K)) < \sum_{n \in S} 1/ n \leq 1$ ?

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.ErdosProblems.«312». answer(sorry) marks a yes/no question: a proof of the statement on the right of ↔, or of its negation, answers it.

theorem erdos_312 :
    answer(sorry) ↔
    ∃ (c : ℝ), 0 < c ∧
      ∀ (K : ℝ), 1 < K →
        ∃ (N₀ : ℕ),
          ∀ (n : ℕ) (a : Fin n → ℕ),
            (n ≥ N₀ ∧ (∑ i : Fin n, (a i : ℝ)⁻¹) > K) →
              ∃ (S : Finset (Fin n)),
                1 - Real.exp (-(c * K)) < (∑ i ∈ S, (a i : ℝ)⁻¹) ∧
                ∑ i ∈ S, (a i : ℝ)⁻¹ ≤ 1

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/312. Report it as a literature claim.
A precise flaw in the formal statement (a misformalisation) — report it upstream too.

References

erdosproblems.com/312

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.