FormalConjectures.Paper.Chvatal

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import FormalConjecturesUtil

Chvátal's Conjecture

References:

A Conjecture in Extremal Combinatorics

Chvátal's Conjecture and Correlation Inequalities


namespace Chvatal


variable {α : Type} [Fintype α] [DecidableEq α] [Nonempty α]

A family F of sets is Decreasing if it is closed under taking subsets.

def Decreasing (F : Finset (Finset α)) : Prop :=
    ∀ A B : Finset α, B ⊆ A → A ∈ F → B ∈ F

A family F of sets is Intersecting if each pair of members has nonempty intersection.

def Intersecting (F : Finset (Finset α)) : Prop :=
    ∀ A ∈ F, ∀ B ∈ F, A ∩ B ≠ ∅

If F is a decreasing family of sets of some finite type α, then there is some element x of α such that the family consisting of all members of F containing x is an intersecting subfamily of F with maximal cardinality.

@[category research open, AMS 5]
theorem declaration uses 'sorry'exists_maximal_star :
    ∀ F : Finset (Finset α), Decreasing F →
        ∃ x : α, ∀ G, G ⊆ F → Intersecting G → G.card ≤ { A ∈ F | x ∈ A }.card := byα:Typeinst✝²:Fintype αinst✝¹:DecidableEq αinst✝:Nonempty α⊢ ∀ (F : Finset (Finset α)), Decreasing F → ∃ x, ∀ G ⊆ F, Intersecting G → G.card ≤ {A ∈ F | x ∈ A}.card
  sorryAll goals completed! 🐙