Erdős Problem 20

Reference: erdosproblems.com/20

def IsSunflowerWithKernel {α : Type} (F : Set (Set α)) (S : Set α) : Prop

A sunflower $F$ with kernel $S$ is a collection of sets in which all possible distinct pairs of sets share the same intersection $S$.

Equations

• IsSunflowerWithKernel F S = F.Pairwise fun (A B : Set α) => A ∩ B = S

def IsSunflower {α : Type} (F : Set (Set α)) : Prop

A sunflower $F$ is a collection of sets in which all possible distinct pairs of sets share the same intersection.

Equations

• IsSunflower F = ∃ (S : Set α), IsSunflowerWithKernel F S

noncomputable def f (n k : ℕ) : ℕ

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.

Equations

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

theorem erdos_20 : ∃ (c : ℕ → ℕ), ∀ (n k : ℕ), f n k < c k ^ n

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