The board / set theory / #593
Erdős Problem #593
OPENdatabase updated 2025-08-31
No proof or disproof is known, and it isn't known whether a finite computation could settle it.
What the community database records, not our verdict. erdosproblems.com no longer shows statuses (why).
**Erdős Problem 593 (\\(500)**: Characterize those finite 3-uniform hypergraphs which appear in every 3-uniform hypergraph of chromatic number \)> \aleph_0\(. The answer is the set of obligatory finite 3-uniform hypergraphs, represented here on the labelled vertex sets `Fin n`. Two-colorability (Property B) is a necessary condition, see `erdos_593.variants.obligatory_implies_two_colorable`, but it is not sufficient: two triples sharing a pair form a 2-colorable hypergraph that is not obligatory, see `erdos_593.variants.common_pair_not_obligatory`. In the graph case (\)r = 2$) the problem is completely solved by Erdős–Galvin–Hajnal [EGH75]: the obligatory graphs are exactly the finite bipartite graphs. A resolution has been claimed by E. Li (arXiv:2606.24882, 2026); at the time of writing erdosproblems.com still lists the problem as open.
Statement as formalized in Formal Conjectures (Google DeepMind, Apache-2.0) · their label: research open · 5 further variants / known partial results in the Lean file. The authoritative wording is on erdosproblems.com/593.Attempts
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A $500-prize problem (#593, characterize the finite triple systems every uncountably-chromatic triple system must contain) plus a companion (#1177, three exact-cardinal questions about avoidance classes) — both already claimed by Eric Li, already flagged by the community as warranting extra scrutiny given his pace of claims. We gave it that scrutiny: cloned the repo, built from source, ran #print axioms on all 8 component theorems plus the unconditional master theorem. Clean.
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