666.lean`

Explanation legend Lean4 Domain Explanation Language (L4DEL)

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L4DEL: Lean4 Domain Explanation Language

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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 666

References:

erdosproblems.com/666
[BDT93] Brouwer, A. E. and Dejter, I. J. and Thomassen, C., Highly symmetric subgraphs of hypercubes. J. Algebraic Combin. (1993), 25-29.
[Ch92] Chung, Fan R. K., Subgraphs of a hypercube containing no small even cycles. J. Graph Theory (1992), 273-286.
[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 Erdos666.erdos_666 Let $Q_n$ be the $n$-dimensional hypercube graph (so that $Q_n$ has $2^n$ vertices and $n2^{n-1}$ edges) sorry line 53 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, for every $\epsilon>0$, if $n$ is sufficiently large, every subgraph of $Q_n$ with $$\geq \epsilon n2^{n-1}$$ many edges contains a $C_6$?
The answer to this problem is no: Chung [Ch92] and Brouwer, Dejter, and Thomassen [BDT93] constructed an edge-partition of $Q_n$ into four subgraphs, each containing no $C_6$.

depends on sorry what this means
Statement uses: Iff, False, Real, LT.lt, OfNat.ofNat, Filter.Eventually, Nat, SimpleGraph, Fin, Bool, LE.le, SimpleGraph.hypercube, HMul.hMul, HPow.hPow, HSub.hSub, Set.ncard, Sym2, SimpleGraph.edgeSet, SimpleGraph.IsContained, SimpleGraph.cycleGraph and 1 more
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
40 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_666 : answer(False) ↔
    ∀ ε : ℝ, 0 < ε → ∀ᶠ n : ℕ in atTop, ∀ H : SimpleGraph (Fin n → Bool), H ≤ hypercube n →
      ε * n * 2 ^ (n - 1 : ℕ) ≤ (H.edgeSet.ncard : ℝ) → (cycleGraph 6 ⊑ H) := by
…
```