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OP-9309OpenGraph theory

Erdős #1035 · graph theory

Is there a constant c>0c>0 such that every graph on 2n2^n vertices with minimum degree >(1−c)2n>(1-c)2^n contains the nn-dimensional hypercube QnQ_n?

MSC 05C · Graph theory

Inferred from the community field; finer classification has not been reviewed.

Source reports OPENAs of 2026-09-08Source & evidence ↓
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Problem statement

Is there a constant c>0c>0 such that every graph on 2n2^n vertices with minimum degree >(1−c)2n>(1-c)2^n contains the nn-dimensional hypercube QnQ_n?

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Formal statement

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Background

Erdős [Er93] says 'if the conjecture is false, two related problems could be asked':

  • Determine or estimate the smallest m>2nm>2^n such that every graph on mm vertices with minimum degree >(1−c)2n>(1-c)2^n contains a QnQ_n, and
  • For which unu_n is it true that every graph on 2n2^n vertices with minimum degree >2n−un>2^n-u_n contains a QnQ_n. See also [576] for the extremal number of edges that guarantee a QnQ_n.

Progress & source assessment

The source labels this entry OPEN in the 2026-09-08 snapshot. Editorial progress notes are reproduced as source commentary, not independently verified results. Prize amounts and Lean badges at the source are not local awards or verification.

References
  1. [1]Thomas Bloom (ed.), Erdős Problems #1035 link
  2. [2]Upstream Lean 4 statement for Erdős #1035 linkExternal statement reference. Not compiled or verified by this platform; upstream files may contain open proof obligations.
  3. [3]Formal Conjectures — Erdős Problem 1035 link

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