Erdős (Erdos) Problem #1191: status in 2026

Erdős Problem #1191

Let A⊂ℕ be an infinite Sidon set. Is it true that liminf_x→ ∞ lvert A∩ [1,x]rvert/x^1/2(log x)^1/2=0? Here x runs over ℕ and the liminf is taken in ℝ≥0∞, so that it is ∞ rather than 0 when the sequence tends to infinity.

Status (as of 2026-11-11): Open according to erdosproblems.com (status last updated 2026-04-04). Erdős offered a prize of $1000.

Current state (tracked automatically)

erdosproblems.com lists Erdős problem #1191 as open (status updated 4 April 2026).

All tracked facts and sources

Fact	Value	Source
Status of Erdős problem #1191 on erdosproblems.com	open 2026-04-04	erdosproblems.com
Formal Conjectures: research statements in the file	2 open, 1 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.

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Cite

@misc{cairn-erdos-1191,
  title        = {Erdős Problem #1191},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/erdos-1191}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-11-12}
}

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Claims

Verified

Disputed

Refuted

On the literature board

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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

erdos_1191.parts.i. Let $A \subset N$ be an infinite Sidon set. Is it true that $x \to \infty lim inf \frac{∣ A \cap [ 1 , x ]∣}{x ^{1/2}} (lo g x)^{1/2} = 0 ?$

Here $x$ runs over $N$ and the $lim inf$ is taken in ℝ≥0∞, so that it is $\infty$ rather than $0$ when the sequence tends to infinity.

erdos_1191.parts.ii. Does there exist an infinite Sidon set $A$ such that $x \to \infty lim inf \frac{∣ A \cap [ 1 , x ]∣}{x ^{1/2}} (lo g x)^{c} > 0$ for some $c > 0$ ?

A positive answer would also give a positive answer to Erdős Problem 39.

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.ErdosProblems.«1191» (2 statements).

theorem erdos_1191.parts.i : ∀ A : Set ℕ, A.Infinite → IsSidon A →
    liminf (fun x : ℕ ↦ ENNReal.ofReal
      ((A ∩ .Icc 1 x).ncard / (x : ℝ) ^ (1 / 2 : ℝ) * log x ^ (1 / 2 : ℝ))) atTop = 0

theorem erdos_1191.parts.ii : ∃ A : Set ℕ, A.Infinite ∧ IsSidon A ∧
    ∃ c > (0 : ℝ), 0 < liminf (fun x : ℕ ↦ ENNReal.ofReal
      ((A ∩ .Icc 1 x).ncard / (x : ℝ) ^ (1 / 2 : ℝ) * log x ^ c)) atTop

What counts as progress

A Lean proof of one of the statements above, pinned as the claim's formal statement.
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/1191. Report it as a literature claim.
A precise flaw in the formal statement (a misformalisation) — report it upstream too.

References

erdosproblems.com/1191
[Er80] Erdős, Paul, A survey of problems in combinatorial number theory. Ann. Discrete Math. (1980), 89-115.
[HaRo66] Halberstam, H. and Roth, K. F., Sequences. Vol. I. (1966), xx+291.

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.