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Level A · Machine-checkable Hard Graph theory P-erdos-86

Erdős Problem #86

Let Q_n be the n-dimensional hypercube graph (so that Q_n has 2^n vertices and n2^n-1 edges). Is it true that every subgraph of Q_n with ≥ (1/2+o(1))n2^n-1 many edges contains a 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-86,
  title        = {Erdős Problem #86},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/erdos-86}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-09-28}
}

Also: CITATION.cff · Atom feed of results

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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

Let be the -dimensional hypercube graph (so that has vertices and edges). Is it true that every subgraph of with many edges contains a ?

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.ErdosProblems.«86». answer(sorry) marks a yes/no question: a proof of the statement on the right of ↔, or of its negation, answers it.

theorem erdos_86 : answer(sorry) ↔
    ∀ ε : ℝ, 0 < ε → ∀ᶠ n : ℕ in atTop, ∀ H : SimpleGraph (Fin n → Bool),
      H ≤ hypercube n →
        (1 / 2 + ε) * n * 2 ^ (n - 1 : ℕ) ≤ (H.edgeSet.ncard : ℝ) →
          cycleGraph 4 ⊑ H

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/86. Report it as a literature claim.
  • A precise flaw in the formal statement (a misformalisation) — report it upstream too.

References

  • erdosproblems.com/86
  • [BHLL14] Balogh, József and Hu, Ping and Lidický, Bernard and Liu, Hong, *Upper bounds on the size of 4- and 6-cycle-free subgraphs of the hypercube*. European J. Combin. (2014), 75-85.
  • [BHN95] Brass, Peter and Harborth, Heiko and Nienborg, Hauke, *On the maximum number of edges in a {}-free subgraph of {}*. J. Graph Theory (1995), 17--23.
  • [Ba12b] R. Baber, Turán densities of hypercubes. arXiv:1201.3587 (2012).
  • [Er91] Erdős, P., *Problems and results in combinatorial analysis and combinatorial number theory*. Graph theory, combinatorics, and applications, Vol. 1 (Kalamazoo, MI, 1988) (1991), 397-406.

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.