FormalConjectures.ErdosProblems.«857»

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import FormalConjecturesUtil

Erdős Problem 857

Reference: erdosproblems.com/857

For fixed n, k, let m(n, k) be minimal such that every family of subsets of [n] of size at least m(n, k) contains a k-sunflower. The problem asks to estimate m(n, k), ideally asymptotically.


open Filter


namespace Erdos857

m(n, k): minimal sunflower-forcing family size in the non-uniform [n] model.

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

Estimate m(n,k), or better give an asymptotic formula.

@[category research open, AMS 5]
theorem declaration uses 'sorry'erdos_857 :
    let f : ℕ → ℕ → ℝ := answer(sorry)
    ∀ k : ℕ, 3 ≤ k → Tendsto (fun n : ℕ => (m n k : ℝ) / f k n) atTop (nhds 1) := by⊢ let f := sorry;
∀ (k : ℕ), 3 ≤ k → Tendsto (fun n => ↑(m n k) / f k n) atTop (nhds 1)
  sorryAll goals completed! 🐙


end Erdos857