Erdős (Erdos) Problem #30: status in 2026

Erdős Problem #30

Is it true that, for every ε > 0, h(N) = sqrt N + O_ε(N^ε)

Status (as of 2026-11-11): Open according to erdosproblems.com (status last updated 2025-08-31). Erdős offered a prize of $1000.

Current state (tracked automatically)

erdosproblems.com lists Erdős problem #30 as open (status updated 31 August 2025).
Formal Conjectures marks the Lean statement erdos_30 as open.

All tracked facts and sources

Fact	Value	Source
Status of Erdős problem #30 on erdosproblems.com	open 2025-08-31	erdosproblems.com
Formal Conjectures: status of `erdos_30`	research open	Formal Conjectures
Formal Conjectures: research statements in the file	2 open, 7 solved	Formal Conjectures

Read twice a day from public sources by deterministic collectors (no AI; the sentences above are filled-in templates): erdosproblems.com (checked 2026-11-12), Formal Conjectures (checked 2026-11-12). How it works.

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-30,
  title        = {Erdős Problem #30},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/erdos-30}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-11-13}
}

Also: CITATION.cff · Atom feed of results

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Claims

Verified

Disputed

Refuted

On the literature board

Nobody has worked on this problem here yet

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What a result looks like: C-16, a newcomer working with an AI model found C(85,24,3) ≤ 76 (previous best 77), and a computer verified it within minutes. A documented dead end counts too.

The problem

The question

Is it true that, for every $ε > 0$ , $h (N) = N + O_{ε} (N^{ε})$

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.ErdosProblems.«30».

theorem erdos_30 :
    ∀ᵉ (ε > 0), (fun N => h N - (N : Real).sqrt) =O[atTop] fun N => (N : ℝ)^(ε : 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/30. Report it as a literature claim.
A precise flaw in the formal statement (a misformalisation) — report it upstream too.

Variants

erdos_30.variants.O_one — A stronger conjecture: is it true that h(N) = sqrt N + O(1)? Erdős thought this was perhaps too optimistic.

References

erdosproblems.com/30
[ErTu41] Erdős, P. and Turán, P., *On a problem of Sidon in additive number theory, and on some related problems*. J. London Math. Soc. 16 (1941), 212-215.
[Li69] Lindström, B., An inequality for $B_{2}$ -sequences. J. Combinatorial Theory 6 (1969), 211-212.
[Si38] Singer, J., *A theorem in finite projective geometry and some applications to number theory*. Trans. Amer. Math. Soc. 43 (1938), 377-385.
[BFR23] Balogh, J., Füredi, Z. and Roy, S., An upper bound on the size of Sidon sets. Amer. Math. Monthly 130 (2023), 437-445.
[OB22] O'Bryant, K., On the size of finite Sidon sets. arXiv:2207.07800 (2022).
[CHO25] Carter, D., Hunter, Z. and O'Bryant, K., On the diameter of finite Sidon sets. Acta Math. Hungar. 175 (2025), 108-126.

See also Ben Green's Open Problem 31 (formalised in FormalConjectures/GreensOpenProblems/31.lean).

Source and licence

Imported from Formal Conjectures (Erdős problems), commit 2f91d251c07f. Statements and descriptions © The Formal Conjectures Authors, Apache License 2.0; reformatted for this page.