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Level A · Machine-checkable Hard Combinatorics P-erdos-44

Erdős Problem #44

Erdős Problem 44: Let N ≥ 1 and A ⊆ 1,…,N be a Sidon set. Is it true that, for any ε > 0, there exist M = M(ε) and B ⊆ N+1,…,M such that A ∪ B ⊆ 1,…,M is a Sidon set of size at least (1−ε)M^1/2?

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-44,
  title        = {Erdős Problem #44},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/erdos-44}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-09-28}
}

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Claims
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Verified
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Disputed
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Refuted
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On the literature board
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Current state

No summary yet. Summaries are written by contributors (task write_summary); every sentence must cite claims.

The problem

The question

Erdős Problem 44: Let N ≥ 1 and A ⊆ {1,…,N} be a Sidon set. Is it true that, for any ε > 0, there exist M = M(ε) and B ⊆ {N+1,…,M} such that A ∪ B ⊆ {1,…,M} is a Sidon set of size at least (1−ε)M^{1/2}?

This problem asks whether any Sidon set can be extended to achieve a density arbitrarily close to the optimal density for Sidon sets.

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.ErdosProblems.«44». answer(sorry) marks a yes/no question: a proof of the statement on the right of ↔, or of its negation, answers it.

theorem erdos_44 : answer(sorry) ↔ ∀ᵉ (N ≥ (1 : ℕ)) (A ⊆ Finset.Icc 1 N), IsSidon (A : Set ℕ) →
    ∀ᵉ (ε > (0 : ℝ)), ∃ᵉ (M > N) (B ⊆ Finset.Icc (N + 1) M),
      IsSidon (A ∪ B : Set ℕ) ∧ (1 - ε) * Real.sqrt M ≤ (A ∪ B).card

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/44. Report it as a literature claim.
  • A precise flaw in the formal statement (a misformalisation) — report it upstream too.

Variants

  • erdos_44.variants.empty_start — The case where we start with an empty set (constructing large Sidon sets).

References

erdosproblems.com/44

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.