FormalConjectures.ErdosProblems.«623»

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import FormalConjecturesUtil

Erdős Problem 623

Reference: erdosproblems.com/623

open Filter Real Setopen scoped Cardinal Ordinalnamespace Erdos623universe u

Let $X$ be a set of cardinality $\aleph_\omega$ and $f$ be a function from the finite subsets of $X$ to $X$ such that $f(A)\not\in A$ for all $A$. Must there exist an infinite $Y\subseteq X$ that is independent - that is, for all finite $B\subset Y$ we have $f(B)\not\in Y$?

@[category research open, AMS 3]
theorem declaration uses 'sorry'erdos_623 : answer(sorry) ↔ ∀ (X : Type u) (hX : #X = ℵ_ ω)
    (f : Finset X → X), (∀ A : Finset X, f A ∉ A) →
    (∃ Y : Set X, Set.Infinite Y ∧ (∀ (B : Finset X), ↑B ⊆ Y → f B ∉ Y)) := by⊢ True ↔
  ∀ (X : Type u),
    #X = ℵ_ ω → ∀ (f : Finset X → X), (∀ (A : Finset X), f A ∉ A) → ∃ Y, Y.Infinite ∧ ∀ (B : Finset X), ↑B ⊆ Y → f B ∉ Y
  sorryAll goals completed! 🐙


-- TODO(firsching): formalize the statement about X < ℵ_ω

end Erdos623