602.lean

/-
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-/

import FormalConjectures.Util.ProblemImports

/-!
# Erdős Problem 602

*Reference:* [erdosproblems.com/602](https://www.erdosproblems.com/602)
-/

namespace Erdos602

open Cardinal

universe u v

variable (ι : Type u) (α : Type v) (A : ι → Set α) (h : ∀ i, #(A i) = ℵ₀)
variable (hij : Pairwise fun i j ↦ (A i ∩ A j).Finite)

/--
**Erdős Problem 602:**
Let $(A_i)$ be a family of sets with $|A_i| = \aleph_0$ for all $i$, such that for any $i \neq j$ we have
$|A_i \cap A_j|$ finite and $\neq 1$.

Is there a $2$-colouring of $\bigcup_i A_i$ such that no $A_i$ is monochromatic?
-/
@[category research open, AMS 03 05]
theorem erdos_602 :
    (∃ (c : α → Fin 2),
    ∀ i, #(c '' {x : α | x ∈ A i}) ≠ 1) ↔ answer(sorry) := by
  sorry

end Erdos602