Erdős Problem #1035 — open problem in graph theory

Erdős Problem #1035

Is there a constant c > 0 such that every graph on 2^n vertices with minimum degree > (1-c) · 2^n contains the n-dimensional hypercube Q_n? This is Erdős's question [Er93, p. 345]. See also [576] for the extremal number of edges that guarantee a Q_n.

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-1035,
  title        = {Erdős Problem #1035},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/erdos-1035}},
  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

Is there a constant $c > 0$ such that every graph on $2^{n}$ vertices with minimum degree $> (1 - c) \cdot 2^{n}$ contains the $n$ -dimensional hypercube $Q_{n}$ ?

This is Erdős's question [Er93, p. 345].

See also [576] for the extremal number of edges that guarantee a $Q_{n}$ .

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.ErdosProblems.«1035». answer(sorry) marks a yes/no question: a proof of the statement on the right of ↔, or of its negation, answers it.

theorem erdos_1035 : answer(sorry) ↔
    ∃ c > 0, ∀ n : ℕ, ∀ (G : SimpleGraph (Fin (2 ^ n))) [DecidableRel G.Adj],
      (∀ v, (G.degree v : ℝ) > (1 - c) * 2 ^ n) →
        (SimpleGraph.hypercube n).IsContained G

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/1035. Report it as a literature claim.
A precise flaw in the formal statement (a misformalisation) — report it upstream too.

References

erdosproblems.com/1035
[Er93] Erdős, Paul, Some of my favorite solved and unsolved problems in graph theory. Quaestiones Math. (1993), 333-350.

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.