Erdős Problem #326 — open problem in combinatorics

Erdős Problem #326

Does there exist A = a_1 < a_2 < ⋯ ⊂ ℕ which is a minimal basis of order 2 (i.e. every large integer is the sum of 2 elements from A, and no proper subset of A has this property), such that lim_k→∞ a_k/k^2 = c for some c ≠ 0? Erdős and Graham conjectured a negative answer to this question [ErGr80].

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-326,
  title        = {Erdős Problem #326},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/erdos-326}},
  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

Does there exist $A = {a_{1} < a_{2} < \dots} \subset N$ which is a minimal basis of order $2$ (i.e. every large integer is the sum of $2$ elements from $A$ , and no proper subset of $A$ has this property), such that $k \to \infty lim \frac{a _{k}}{k ^{2}} = c$ for some $c \neq = 0$ ?

Erdős and Graham conjectured a negative answer to this question [ErGr80].

"Minimal basis of order $2$ " is formalised as Minimal for the predicate Set.IsAsymptoticAddBasisOfOrder · 2 on sets of naturals ordered by inclusion.

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.ErdosProblems.«326». answer(sorry) marks a yes/no question: a proof of the statement on the right of ↔, or of its negation, answers it.

theorem erdos_326 : answer(sorry) ↔ ∃ (a : ℕ → ℕ), StrictMono a ∧
    Minimal (fun A : Set ℕ ↦ A.IsAsymptoticAddBasisOfOrder 2) (Set.range a) ∧
      ∃ (c : ℝ), c ≠ 0 ∧ Tendsto (fun n ↦ (a n : ℝ) / n ^ 2) atTop (𝓝 c)

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/326. Report it as a literature claim.
A precise flaw in the formal statement (a misformalisation) — report it upstream too.

References

erdosproblems.com/326
[ErGr80] Erdős, P. and Graham, R. L., *Old and new problems and results in combinatorial number theory*. Monographies de L'Enseignement Mathématique (1980), p. 47.
[Ca57] Cassels, J. W. S., Über Basen der natürlichen Zahlenreihe. Abh. Math. Sem. Univ. Hamburg (1957), 247-257.

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.