Erdős (Erdos) Problem #86: status in 2026

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?

Status (as of 2026-11-11): Open according to erdosproblems.com (status last updated 2025-08-31). Erdős offered a prize of $100.

Current state (tracked automatically)

erdosproblems.com lists Erdős problem #86 as open (status updated 31 August 2025).
Formal Conjectures marks the Lean statement erdos_86 as open.

All tracked facts and sources

Fact	Value	Source
Status of Erdős problem #86 on erdosproblems.com	open 2025-08-31	erdosproblems.com
Formal Conjectures: status of `erdos_86`	research open	Formal Conjectures
Formal Conjectures: research statements in the file	1 open, 0 solved	Formal Conjectures

Read twice a day from public sources by deterministic collectors (no AI; the sentences above are filled-in templates): erdosproblems.com (checked 2026-11-12), Formal Conjectures (checked 2026-11-12). How it works.

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 2026-11-12}
}

Also: CITATION.cff · Atom feed of results

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Claims

Verified

Disputed

Refuted

On the literature board

Nobody has worked on this problem here yet

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What a result looks like: C-16, a newcomer working with an AI model found C(85,24,3) ≤ 76 (previous best 77), and a computer verified it within minutes. A documented dead end counts too.

The problem

The question

Let $Q_{n}$ be the $n$ -dimensional hypercube graph (so that $Q_{n}$ has $2^{n}$ vertices and $n 2^{n - 1}$ edges). Is it true that every subgraph of $Q_{n}$ with $\geq (\frac{1}{2} + o (1)) n 2^{n - 1}$ many edges contains a $C_{4}$ ?

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.ErdosProblems.«86».

theorem erdos_86 :
    ∀ 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 { $C_{4}$ }-free subgraph of { $Q_{n}$ }*. 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 2f91d251c07f. Statements and descriptions © The Formal Conjectures Authors, Apache License 2.0; reformatted for this page.