FormalConjectures.ErdosProblems.«701»

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import FormalConjecturesUtil

Erdős Problem 701

Reference: erdosproblems.com/701


namespace Erdos701


open Cardinal

Let $\mathcal{F}$ be a family of sets closed under taking subsets (i.e. if $B\subseteq A\in\mathcal{F}$ then $B\in \mathcal{F}$). There exists some element $x$ such that whenever $\mathcal{F}'\subseteq \mathcal{F}$ is an intersecting subfamily we have $$\lvert \mathcal{F}'\rvert \leq \lvert { A\in \mathcal{F} : x\in A}\rvert.$$

@[category research open, AMS 5]
theorem declaration uses 'sorry'erdos_701 : answer(sorry) ↔ ∀ {X : Type} [Nonempty X] [Fintype X],
    ∀ (F : Set (Set X)), IsLowerSet F →
      ∃ x : X, ∀ᵉ (F' ⊆ F),
        F'.Intersecting →
          (#F' ≤ #{ A : Set X | A ∈ F ∧ x ∈ A }) := by⊢ True ↔
  ∀ {X : Type} [Nonempty X] [Fintype X] (F : Set (Set X)),
    IsLowerSet F → ∃ x, ∀ F' ⊆ F, F'.Intersecting → #↑F' ≤ #↑{A | A ∈ F ∧ x ∈ A}
  sorryAll goals completed! 🐙


end Erdos701