FormalConjectures.ErdosProblems.«447»

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import FormalConjecturesUtil

Erdős Problem 447

References:

erdosproblems.com/447

[Er65b] Erdős, Paul, Some recent advances and current problems in number theory. Lectures on Modern Mathematics, Vol. III (1965), 196-244.

[Kl71] Kleitman, Daniel, Collections of subsets containing no two sets and their union. Proceedings of the LA Meeting AMS (1971), 153-155.

open Filternamespace Erdos447

The largest size of a union-free collection $\mathcal{F}$ of subsets of $[n]$.

noncomputable def maxUnionFree (n : ℕ) : ℕ :=
  sSup { k | ∃ F : Finset (Finset (Fin n)), F.UnionFree ∧ F.card = k }

How large can a union-free collection $\mathcal{F}$ of subsets of $[n]$ be? By union-free we mean there are no solutions to $A\cup B=C$ with distinct $A,B,C\in \mathcal{F}$. Must $\lvert \mathcal{F}\rvert =o(2^n)$?

In [Er65b] Erdős reported that the estimate $\lvert \mathcal{F}\rvert=o(2^n)$ was proved in unpublished work by Sárközy and Szemerédi.

@[category research solved, AMS 5]
theorem erdos_447.parts.i : answer(True) ↔
    (fun n : ℕ => (maxUnionFree n : ℝ)) =o[atTop] (fun n : ℕ => (2 : ℝ) ^ n) := by⊢ True ↔ (fun n ↦ ↑(maxUnionFree n)) =o[atTop] fun n ↦ 2 ^ n
  sorryAll goals completed! 🐙

How large can a union-free collection $\mathcal{F}$ of subsets of $[n]$ be? By union-free we mean there are no solutions to $A\cup B=C$ with distinct $A,B,C\in \mathcal{F}$. Perhaps even $$\lvert \mathcal{F}\rvert <(1+o(1))\binom{n}{\lfloor n/2\rfloor}?$$

Solved by Kleitman [Kl71], who proved $$\lvert \mathcal{F}\rvert <(1+o(1))\binom{n}{\lfloor n/2\rfloor}.$$

@[category research solved, AMS 5, formal_proof using lean4 at "https://github.com/plby/lean-proofs/blob/main/src/v4.29.1/ErdosProblems/Erdos447.lean"]
theorem erdos_447.parts.ii : answer(True) ↔
    ∃ c : ℕ → ℝ, (c =o[atTop] (1 : ℕ → ℝ)) ∧ ∀ᶠ n : ℕ in atTop,
      (maxUnionFree n : ℝ) < (1 + c n) * (n.choose (n / 2) : ℝ) := by⊢ True ↔ ∃ c, c =o[atTop] 1 ∧ ∀ᶠ (n : ℕ) in atTop, ↑(maxUnionFree n) < (1 + c n) * ↑(n.choose (n / 2))
  sorryAll goals completed! 🐙

end Erdos447