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.

Graph analogue — bipartite graphs are obligatory (Erdős–Galvin–Hajnal [EGH75]): For the 2-uniform (graph) case, a graph of chromatic cardinal $> \aleph_0$ must contain all finite bipartite graphs. Specifically, for every finite bipartite graph F and every graph G with chromatic cardinal $> \aleph_0$, there is a graph embedding from F into G.

This uses Nonempty (F ↪g G) (graph embedding), aligned with the injective vertex map used in the hypergraph Appears definition.