A finite triple system is obligatory exactly when (after deleting isolated vertices) it is linear, every hyperedge-node of its Levi graph has a bridge, and all Berge cycles are even; otherwise it is avoidable at every uncountable chromatic number.
Reiher,Graphs of large girth, preprint, 2024, arXiv:2403.13571
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A Resolution of Erd\H{o}s Problems 593 and 1177: Obligatory Triple Systems and Exact Spectra
A finite triple system is obligatory exactly when (after deleting isolated vertices) it is linear, every hyperedge-node of its Levi graph has a bridge, and all Berge cycles are even; otherwise it is avoidable at every uncountable chromatic number.