Erdős (Erdos) Problem #1194: status in 2026

Erdős Problem #1194

Let A⊂ℕ be such that every integer n≥ 1 can be written uniquely as a_n-b_n for some a_n,b_n∈ A. How fast must a_n/n increase? A question of Erdős [Er80, p. 100]. Here a_n is the larger element of the representation of n, not the nth element of A.

Status (as of 2026-11-11): Open according to erdosproblems.com (status last updated 2026-04-04).

Current state (tracked automatically)

erdosproblems.com lists Erdős problem #1194 as open (status updated 4 April 2026).
Formal Conjectures marks the Lean statement erdos_1194 as open.

All tracked facts and sources

Fact	Value	Source
Status of Erdős problem #1194 on erdosproblems.com	open 2026-04-04	erdosproblems.com
Formal Conjectures: status of `erdos_1194`	research open	Formal Conjectures
Formal Conjectures: research statements in the file	1 open, 3 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-1194,
  title        = {Erdős Problem #1194},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/erdos-1194}},
  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

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

Let $A \subset N$ be such that every integer $n \geq 1$ can be written uniquely as $a_{n} - b_{n}$ for some $a_{n}, b_{n} \in A$ . How fast must $a_{n} / n$ increase?

A question of Erdős [Er80, p. 100]. Here $a_{n}$ is the larger element of the representation of $n$ , not the $n$ th element of $A$ . We read "must increase" as: the answer is the set of non-decreasing $G$ such that every such $A$ has $a_{n} \geq c n G (n)$ for infinitely many $n$ , for some $c > 0$ depending on $A$ .

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.ErdosProblems.«1194».

theorem erdos_1194 :
    {G : ℕ → ℝ | Monotone G ∧ ∀ A : Set ℕ, IsPerfectDifferenceSetInt A →
      ∃ c > (0 : ℝ), HasLargeElement A (fun n ↦ c * G n * n)} = erdos_1194.answer

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/1194. Report it as a literature claim.
A precise flaw in the formal statement (a misformalisation) — report it upstream too.

References

erdosproblems.com/1194
[Er80] Erdős, Paul, A survey of problems in combinatorial number theory. Ann. Discrete Math. 6 (1980), 89-115. PDF
[HaRo66] Halberstam, H. and Roth, K. F., Sequences. Vol. I. Oxford University Press, Oxford (1966), xx+291.
[Le04] Lev, Vsevolod F., Reconstructing integer sets from their representation functions. Electron. J. Combin. 11 (2004), Research Paper 78, 6 pp. Link

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.