Erdős Problem #883 — open problem in graph theory

Erdős Problem #883

For A⊆ 1,…,n let G(A) be the graph with vertex set A, where two integers are joined by an edge if they are coprime. Is it true that if |A| > ⌊ n/2 ⌋ + ⌊ n/3 ⌋ - ⌊ n/6 ⌋ then G(A) contains all odd cycles of length ≤ n/3 + 1? A problem of Erdős and Sárközy [ErSa97].

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-883,
  title        = {Erdős Problem #883},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/erdos-883}},
  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

For $A \subseteq {1, \dots, n}$ let $G (A)$ be the graph with vertex set $A$ , where two integers are joined by an edge if they are coprime.

Is it true that if $∣ A ∣ > ⌊ n /2 ⌋ + ⌊ n /3 ⌋ - ⌊ n /6 ⌋$ then $G (A)$ contains all odd cycles of length $\leq n /3 + 1$ ?

A problem of Erdős and Sárközy [ErSa97].

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.ErdosProblems.«883». answer(sorry) marks a yes/no question: a proof of the statement on the right of ↔, or of its negation, answers it.

theorem erdos_883.parts.i : answer(sorry) ↔
    ∀ (n : ℕ) (A : Finset ℕ),
      A ⊆ Finset.Icc 1 n →
      n / 2 + n / 3 - n / 6 < A.card →
      ∀ l : ℕ, Odd l → 3 ≤ l → l ≤ n / 3 + 1 →
        l ∈ (coprimeGraph.induce (A : Set ℕ)).oddCycleLengths

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/883. Report it as a literature claim.
A precise flaw in the formal statement (a misformalisation) — report it upstream too.

References

erdosproblems.com/883
[ErSa97] Erdős, P. and Sárközy, G. N., On cycles in the coprime graph of integers. Electron. J. Combin. (1997), Research Paper 8.
[Sa99] Sárközy, G. N., Complete tripartite subgraphs in the coprime graph of integers. Discrete Math. (1999), 227-238.

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.