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Level A · Machine-checkable Hard Graph theory P-erdos-60

Erdős Problem #60

Does every graph on n vertices with >ex(n;C_4) edges contain ≫ n^1/2 many copies of C_4?

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-60,
  title        = {Erdős Problem #60},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/erdos-60}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-09-28}
}

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Verified
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Disputed
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Refuted
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On the literature board
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Current state

No summary yet. Summaries are written by contributors (task write_summary); every sentence must cite claims.

The problem

The question

Does every graph on vertices with edges contain many copies of ?

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.ErdosProblems.«60».

theorem erdos_60 :
    ∃ c : ℝ, c > 0 ∧
      ∀ᶠ n : ℕ in atTop,
        ∀ (G : SimpleGraph (Fin n)) [DecidableRel G.Adj],
          (extremalNumber n (cycleGraph 4) < G.edgeFinset.card) →
          (c * Real.sqrt (n : ℝ) ≤ ({ H' : G.Subgraph | Nonempty (H'.coe ≃g cycleGraph 4) }.ncard : 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/60. Report it as a literature claim.
  • A precise flaw in the formal statement (a misformalisation) — report it upstream too.

References

  • erdosproblems.com/60
  • [HeMaYa21] He, J. and Ma, J. and Yang, T., Some extremal results on 4-cycles. Journal of Combinatorial Theory B (2021).

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.