FormalConjectures.ErdosProblems.«1023»

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import FormalConjecturesUtil

Erdős Problem 1023

References:

erdosproblems.com/1023

[Er71, p.105] Erdős, P., Some unsolved problems in graph theory and combinatorial analysis. Combinatorial Mathematics and its Applications (Proc. Conf., Oxford, 1969) (1971), 97-109.

open Filteropen scoped Asymptoticsnamespace Erdos1023

F n is the maximal size of a family of subsets of ${1,\ldots,n}$ such that no set in this family is the union of other members of the family.

noncomputable def F (n : ℕ) : ℕ :=
  sSup {m | ∃ S ⊆ (Finset.Icc 1 n).powerset, S.SubfamilyUnionFree ∧ S.card = m}

Let $F(n)$ be the maximal size of a family of subsets of ${1,\ldots,n}$ such that no set in this family is the union of other members of the family. Is it true that there is a constant $c>0$ such that $$F(n)\sim c \frac{2^n}{n^{1/2}}?$$

Hunter observes in the comments that this follows from the solution to [447], which implies $F(n)\sim \binom{n}{n/2}$.

@[category research solved, AMS 5, formal_proof using lean4 at "https://github.com/plby/lean-proofs/blob/main/src/v4.29.1/ErdosProblems/Erdos1023.lean"]
theorem erdos_1023 : answer(True) ↔
    ∃ c : ℝ, 0 < c ∧
      ((fun n : ℕ => (F n : ℝ)) ~[atTop]
        (fun n : ℕ => c * 2 ^ n / (n : ℝ) ^ (1 / 2 : ℝ))) := by⊢ True ↔ ∃ c, 0 < c ∧ (fun n ↦ ↑(F n)) ~[atTop] fun n ↦ c * 2 ^ n / ↑n ^ (1 / 2)
  sorryAll goals completed! 🐙

Erdős and Kleitman proved in unpublished work that $$F(n)\asymp \frac{2^n}{n^{1/2}}.$$ ([Er71] has an exponent of $3/2$, but this is presumably a typo.)

@[category research solved, AMS 5]
theorem erdos_1023.variants.erdos_kleitman :
    (fun n : ℕ => (F n : ℝ)) =Θ[atTop] (fun n : ℕ => (2 : ℝ) ^ n / (n : ℝ) ^ (1 / 2 : ℝ)) := by⊢ (fun n ↦ ↑(F n)) =Θ[atTop] fun n ↦ 2 ^ n / ↑n ^ (1 / 2)
  sorryAll goals completed! 🐙

Hunter observes in the comments that this follows from the solution to [447], which implies $F(n)\sim \binom{n}{n/2}$.

@[category research solved, AMS 5]
theorem erdos_1023.variants.hunter :
    (fun n : ℕ => (F n : ℝ)) ~[atTop] (fun n : ℕ => (n.choose (n / 2) : ℝ)) := by⊢ (fun n ↦ ↑(F n)) ~[atTop] fun n ↦ ↑(n.choose (n / 2))
  sorryAll goals completed! 🐙

end Erdos1023