FormalConjecturesForMathlib.Combinatorics.SetFamily.PropertyB

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module


public import Mathlib.Data.Finset.Basic
public import Mathlib.Data.Fin.Basic

@[expose] public section

Property B

A family of sets has property B when the ground set admits a 2-colouring under which no member of the family is monochromatic. For 3-uniform hypergraphs this is ThreeUniformHypergraph.IsTwoColorable.

namespace Finset

A family F of finite sets has property B (equivalently, chromatic number 2) if there is a 2-colouring of the ground set under which no member of F is monochromatic.

def HasPropertyB {α : Type*} (F : Finset (Finset α)) : Prop :=
  ∃ f : α → Fin 2, ∀ A ∈ F, ∃ x ∈ A, ∃ y ∈ A, f x ≠ f y

end Finset