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import FormalConjectures.Wikipedia.DedekindNumber
import FormalConjecturesUtilErdős Problem 497
References:
[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.
open Filternamespace Erdos497How 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 $$2^{(1+o(1))\binom{n}{\lfloor n/2\rfloor}}.$$
@[category research solved, AMS 5 6, formal_proof using lean4 at "https://github.com/plby/lean-proofs/blob/1d7b3f00780b85ed0462e79a1cd5650ee9055655/src/v4.29.1/ErdosProblems/Erdos497.lean"]
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)) := ⊢ ∃ o, ∃ (_ : o =o[atTop] 1), ∀ (n : ℕ), ↑(DedekindNumber.M' n) = 2 ^ ((1 + o n) * ↑(n.choose (n / 2)))
All goals completed! 🐙end Erdos497