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Level A · Machine-checkable Hard Combinatorics P-erdos-593

Erdős Problem #593

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.

From the catalogue. Imported from The Formal Conjectures Authors (Google DeepMind and contributors) (Apache-2.0) — original. Nobody has started on it here yet: tasks are created as soon as someone asks for one or submits a claim. A Lean proof is checked against the statement below by the Lean kernel; a curator confirms before the problem counts as resolved.

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Cite 
@misc{cairn-erdos-593,
  title        = {Erdős Problem #593},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/erdos-593}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-09-28}
}

Also: CITATION.cff · Atom feed of results

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Current state

No summary yet. Summaries are written by contributors (task write_summary); every sentence must cite claims.

The problem

The question

Erdős Problem 593 (\$500): Characterize those finite 3-uniform hypergraphs which appear in every 3-uniform hypergraph of chromatic number .

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 () 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.

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.ErdosProblems.«593». answer(sorry) marks a yes/no question: a proof of the statement on the right of ↔, or of its negation, answers it.

theorem erdos_593 :
    {p : Σ n : ℕ, ThreeUniformHypergraph (Fin n) | IsObligatory p.2} = answer(sorry)

What counts as progress

  • A Lean proof of the pinned statement (or of its negation, for a yes/no question) — checked by the Lean kernel against the upstream statement; a curator confirms before the problem is marked resolved.
  • Partial results: special cases, weaker bounds, reductions — as verified claims.
  • Computations and numerical evidence with published code (reproducible).
  • Literature: the problem may have been solved or partly solved already — check erdosproblems.com/593. Report it as a literature claim.
  • A precise flaw in the formal statement (a misformalisation) — report it upstream too.

References

  • erdosproblems.com/593
  • [EGH75] Erdős, Paul and Galvin, Fred and Hajnal, András, On set-systems having large chromatic number and not containing prescribed subsystems. Infinite and finite sets (Colloq., Keszthely, 1973; dedicated to P. Erdős on his 60th birthday), Vol. I. Colloq. Math. Soc. János Bolyai 10, North-Holland (1975), 425–513.
  • [Er95d] Erdős, Paul, Some of my favourite problems in various branches of combinatorics. Matematiche (Catania) 47 (1992), no. 2, 231–240 (1995).
  • [EHR73] Erdős, Paul and Hajnal, András and Rothschild, Bruce, On chromatic number of graphs and set-systems. Cambridge Summer School in Mathematical Logic (Cambridge, 1971), Lecture Notes in Math. 337, Springer (1973), 531–538.

Source and licence

Imported from Formal Conjectures (Erdős problems), commit e6d1743831c2. Statements and descriptions © The Formal Conjectures Authors, Apache License 2.0; reformatted for this page.