Erdős Problem #773 — open problem in number theory

Erdős Problem #773

What is the size of the largest Sidon subset A⊆1,2^2,…,N^2? Is it N^1-o(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-773,
  title        = {Erdős Problem #773},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/erdos-773}},
  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

What is the size of the largest Sidon subset $A \subseteq {1, 2^{2}, \dots, N^{2}}$ ? Is it $N^{1 - o (1)}$ ?

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.ErdosProblems.«773». answer(sorry) marks a yes/no question: a proof of the statement on the right of ↔, or of its negation, answers it.

theorem erdos_773 : answer(sorry) ↔
    ∀ ε > (0 : ℝ), ∀ᶠ N : ℕ in atTop,
      (N : ℝ) ^ (1 - ε) ≤
        (Finset.maxSidonSubsetCard
          (Finset.image (fun n : ℕ => n ^ 2) (Finset.Icc 1 N)) : ℝ)

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/773. Report it as a literature claim.
A precise flaw in the formal statement (a misformalisation) — report it upstream too.

References

erdosproblems.com/773
[AlEr85] Alon, Noga and Erdős, P., An application of graph theory to additive number theory. European J. Combin. (1985), 201-203.
[Er80] Erdős, Paul, A survey of problems in combinatorial number theory. Ann. Discrete Math. (1980), 89-115.
[LeTh95] Lefmann, Hanno and Thiele, Torsten, Point sets with distinct distances. Combinatorica (1995), 379--408.

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.