86.lean`

Explanation legend Lean4 Domain Explanation Language (L4DEL)

L4DEL explanations are a separate layer. They do not change the Lean source. Intuition is explanatory prose; numbered items are a reading path; sources link the wording back to evidence.

L4DEL: Lean4 Domain Explanation Language

Annotation provenance. This overlay contains 10 curated AI-written annotations and 7190 mechanical source-derived annotations. They have not been human-reviewed as explanations; treat them as unverified and not mathematical authority. Retained generation history: The 71 original curated annotations were produced in a Claude Sonnet 5.5 session at medium effort; Claude Opus 5.5 is recorded in that session as the advisor model, not the direct author. The bulk source-derived expansion was generated mechanically from pinned source comments by tooling developed in a GPT-5.6 Sol session at ultra reasoning; GPT-5.6 Sol did not originate the quoted source prose. On 2026-10-17 the mechanically generated source-comment annotations of the Formal Conjectures, Carleson, FLT, PFR, CompFiles, Mathematics in Lean and Theorem Proving in Lean overlays were regenerated by the same generator after three corrections made in a Claude Sonnet 5.5 session: titles no longer cut a sentence inside mathematics or end on a dangling word, workflow markers are removed from the quoted prose, and declaration names follow the scanner's namespace resolution. The boilerplate step line was dropped, and comments that were only a row of dashes no longer yield annotations. No new prose was written and no human review is claimed.

Credit: The upstream authors and contributors retain credit for the project and mathematics. lean4-explainer only renders their pinned source and adds the separately disclosed explanation layer.

Project sources: Pinned project documentation, Formal Conjectures paper, arXiv v1

1 declarations, 1 contain sorry, 0 declared as axioms

Erdős Problem 86

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.

Imports (1)

FormalConjecturesUtil

theorem Erdos86.erdos_86 Let $Q_n$ be the $n$-dimensional hypercube graph (so that $Q_n$ has $2^n$ vertices and $n2^{n-1}$ edges) sorry line 45 source
Source documentation
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 $$\geq \left(\frac{1}{2}+o(1)\right)n2^{n-1}$$ many edges contains a $C_4$?

depends on sorry what this means
Statement uses: Iff, True, Real, LT.lt, OfNat.ofNat, Filter.Eventually, Nat, SimpleGraph, Fin, Bool, LE.le, SimpleGraph.hypercube, HMul.hMul, HAdd.hAdd, HDiv.hDiv, HPow.hPow, HSub.hSub, Set.ncard, Sym2, SimpleGraph.edgeSet and 3 more
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
54 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
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 := by
…
```