Erdos Problem #593 asks which finite triple systems occur in every uncountably chromatic triple system. The answer is exactly the class generated from private-vertex expansions of finite bipartite graphs by finite disjoint unions and one-point amalgamations. Problem #1177 on exact spectra is settled alongside it.

Vertices must be uncountable: Every 3-uniform hypergraph with chromatic cardinal $> \aleph_0$ must have an uncountable vertex set.

Proof: If V is countable, there exists an injection φ : V → ℕ. Using distinct natural numbers as colors gives a proper coloring, so $\chi(H) \leq \#\mathbb{N} = \aleph_0$, contradicting $\chi(H) > \aleph_0$.