FormalConjectures.ErdosProblems.«20»

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import FormalConjecturesUtil

Erdős Problem 20

References:

erdosproblems.com/20

Wikipedia

[ErRa60] Erdős, Paul and Rado, Richard. Intersection theorems for systems of sets. J. London Math. Soc. 35 (1960), 85--90.

namespace Erdos20

Let $f(n,k)$ be minimal such that every $F$ family of $n$-uniform sets with $|F| \ge f(n,k)$ contains a $k$-sunflower.

noncomputable def f (n k : ℕ) : ℕ :=
  sInf {m | ∀ {α : Type}, ∀ (F : Set (Set α)),
    ((∀ f ∈ F, f.ncard = n) ∧ m ≤ F.ncard) → ∃ S ⊆ F, S.ncard = k ∧ IsSunflower S}


@[category test, AMS 5]
theorem f_0_1 : f 0 1 = 1 := by⊢ f 0 1 = 1
  refine IsLeast.csInf_eq ⟨fun F hF ↦ ?_, fun n hn ↦ n.pos_of_ne_zero fun hn₀ ↦ ?_⟩refine_1α✝:TypeF:Set (Set α✝)hF:(∀ f ∈ F, f.ncard = 0) ∧ 1 ≤ F.ncard⊢ ∃ S ⊆ F, S.ncard = 1 ∧ IsSunflower Srefine_2n:ℕhn:n ∈ {m | ∀ {α : Type} (F : Set (Set α)), (∀ f ∈ F, f.ncard = 0) ∧ m ≤ F.ncard → ∃ S ⊆ F, S.ncard = 1 ∧ IsSunflower S}hn₀:n = 0⊢ False
  ·refine_1α✝:TypeF:Set (Set α✝)hF:(∀ f ∈ F, f.ncard = 0) ∧ 1 ≤ F.ncard⊢ ∃ S ⊆ F, S.ncard = 1 ∧ IsSunflower S obtain ⟨A, hA⟩ := F.nonempty_of_ncard_ne_zero (byα✝:TypeF:Set (Set α✝)hF:(∀ f ∈ F, f.ncard = 0) ∧ 1 ≤ F.ncard⊢ F.ncard ≠ 0 omegaAll goals completed! 🐙)
    exact ⟨{A}, byα✝:TypeF:Set (Set α✝)hF:(∀ f ∈ F, f.ncard = 0) ∧ 1 ≤ F.ncardA:Set α✝hA:A ∈ F⊢ {A} ⊆ F ∧ {A}.ncard = 1 ∧ IsSunflower {A} simpa using ⟨hA, isSunflower_singleton _⟩All goals completed! 🐙⟩
  ·refine_2n:ℕhn:n ∈ {m | ∀ {α : Type} (F : Set (Set α)), (∀ f ∈ F, f.ncard = 0) ∧ m ≤ F.ncard → ∃ S ⊆ F, S.ncard = 1 ∧ IsSunflower S}hn₀:n = 0⊢ False obtain ⟨S, hS⟩ := (hn (α := ℕ) {} (byn:ℕhn:n ∈ {m | ∀ {α : Type} (F : Set (Set α)), (∀ f ∈ F, f.ncard = 0) ∧ m ≤ F.ncard → ∃ S ⊆ F, S.ncard = 1 ∧ IsSunflower S}hn₀:n = 0⊢ (∀ f ∈ ∅, f.ncard = 0) ∧ n ≤ ∅.ncard simpaAll goals completed! 🐙))
    simp_all [bot_unique hS.1]All goals completed! 🐙

Is it true that $f(n,k) < c_k^n$ for some constant $c_k>0$ and for all $n > 0$?

@[category research open, AMS 5]
theorem declaration uses 'sorry'erdos_20 : answer(sorry) ↔ ∃ (c : ℕ → ℕ), ∀ n k, n > 0 → f n k < (c k) ^ n := by⊢ True ↔ ∃ c, ∀ (n k : ℕ), n > 0 → f n k < c k ^ n
  sorryAll goals completed! 🐙

Erdős and Rado [ErRa60] proved the factorial upper bound for the $k$-sunflower threshold: any family of $n$-uniform sets with more than $(k-1)^n , n!$ members contains a $k$-sunflower, hence $f(n,k) \le (k-1)^n , n! + 1$.

@[category research solved, AMS 5]
theorem declaration uses 'sorry'erdos_20.variants.erdos_rado_bound :
    ∀ n k, n > 0 → 2 ≤ k → f n k ≤ (k - 1) ^ n * n.factorial + 1 := by⊢ ∀ (n k : ℕ), n > 0 → 2 ≤ k → f n k ≤ (k - 1) ^ n * n.factorial + 1
  sorryAll goals completed! 🐙


end Erdos20