497.lean`

Explanation legend Lean4 Domain Explanation Language (L4DEL)

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L4DEL: Lean4 Domain Explanation Language

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

1 declarations, 1 contain sorry, 0 declared as axioms

Erdős Problem 497

References:

erdosproblems.com/497
[Er61] Erdős, Paul, Some unsolved problems. Magyar Tud. Akad. Mat. Kutató Int. Közl. (1961), 221-254.
[Kl69] Kleitman, Daniel, On Dedekind's problem: The number of monotone Boolean functions. Proc. Amer. Math. Soc. (1969), 677-682.

Imports (2)

theorem Erdos497.erdos_497 How many antichains in $[n]$ are there? sorry line 49 source
Source documentation
How many antichains in $[n]$ are there? That is, how many families of subsets of $[n]$ are there such that, if $\mathcal{F}$ is such a family and $A,B\in \mathcal{F}$, then $A\not\subseteq B$?
Sperner's theorem states that $\lvert \mathcal{F}\rvert \leq \binom{n}{\lfloor n/2\rfloor}$. This is also known as Dedekind's problem. Resolved by Kleitman [Kl69], who proved that the number of such families is $$2^{(1+o(1))\binom{n}{\lfloor n/2\rfloor}}.$$

depends on sorry what this means
Statement uses: Exists, Nat, Real, Asymptotics.IsLittleO, Filter.atTop, OfNat.ofNat, Eq, DedekindNumber.M', HPow.hPow, HMul.hMul, HAdd.hAdd, Nat.choose, HDiv.hDiv
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
38 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_497 :
    ∃ (o : ℕ → ℝ) (_ : o =o[atTop] (1 : ℕ → ℝ)),
      ∀ n : ℕ, (DedekindNumber.M' n : ℝ) = 2 ^ ((1 + o n) * (n.choose (n / 2) : ℝ)) := by
…
```
Mentions: DedekindNumber.M'