**Erdős Problem 598:**
Let $m$ be an infinite cardinal and $\kappa$ be the successor cardinal of $2^{\aleph_0}$.
Can one colour the countable subsets of $m$ using $\kappa$ many colours so that every
$X \subseteq m$ with $|X| = \kappa$ contains subsets of all possible colours?
Actions
Submit a Proof
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theorem erdos_598 : answer(sorry) ↔
∃ c : { s : Set m // s.Countable } → κ.out,
∀ X : Set m, #X = κ →
c '' { s : { sub : Set m // sub.Countable } | s.1 ⊆ X } = Set.univ
