Problem 1177

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claims/: The 1 claim page of Problem 1177, one per claimant's result; the problem's standing derives from them.

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Statement. Let $G$ be a finite $3$-uniform hypergraph, and let $F_G(\kappa)$ denote the collection of $3$-uniform hypergraphs with chromatic number $\kappa$ not containing $G$.

If $F_G(\aleph_1)$ is not empty then there exists $X\in F_G(\aleph_1)$ of cardinality at most $2^{2^{\aleph_0}}$.

If both $F_G(\aleph_1)$ and $F_H(\aleph_1)$ are non-empty then $F_G(\aleph_1)\cap F_H(\aleph_1)$ is non-empty.

If $\kappa,\lambda$ are uncountable cardinals and $F_G(\kappa)$ is non-empty then $F_G(\lambda)$ is non-empty.

Status. Open. The site labels the problem OPEN (page last edited 23 January 2026); the one result claimed against the problem is recorded on its claim page, and the standing in the frontmatter is derived from it.

Source. erdosproblems.com/1177, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #1177, https://www.erdosproblems.com/1177.

References.

• [Va99] Some of Paul's favorite problems, booklet for the conference "Paul Erdős and his mathematics", Budapest, July 1999; item 7.94, attributed to Erdős, Galvin and Hajnal, whose parts (a), (b) and (c) are the three assertions, with one difference: the booklet phrases (a) and (b) for uncountably chromatic triple systems, where the site asks for chromatic number exactly $\aleph_1$; Li's corollary proves the exact version and notes that it implies the booklet's. Library home: various_1999_some_pauls_favorite_problems.

Formalization. Statement in formal-conjectures.

Current assessment

Search scope, 2026-10-06: the site's problem page, discussion thread and proof-claims tab, Li's arXiv preprint and the README of the claimant's repository, and the booklet's item 7.94. Not searched: arXiv beyond the preprint, zbMATH, MathSciNet and X. The proofs of the claimed resolution have not been independently checked.

Claims. One result is claimed from outside the project. Li's exact spectra (arXiv preprint of 2026-06-23, found with GPT-5.5 Pro, with the author's own Lean development) answers the three assertions yes, no and yes: the first and third from a dichotomy that every finite triple system outside an explicit class is avoided by triple systems of every uncountable chromatic number, with a witness of size at most $2^{2^{\aleph_0}}$ at $\aleph_1$, and the second refuted by two triples sharing a pair together with the loose $7$-cycle. The site has not accepted it, the corpus has not built its Lean, and the claim stays claimed, so the problem's standing is claimed with the claim answered, a mixed outcome: the second assertion fails while the first and third hold. The same manuscript claims the classification asked by Problem 593.

Known Results

The claimed resolution is on Li's claim page; no other result on the three assertions is compiled.

Linked library material

These entries are derived from explicit links on library pages. They are navigation only and do not by themselves record mathematical progress.

• various_1999_some_pauls_favorite_problems
• erdos_1975_set_systems_having_large_chromatic_number
• erdos_1975_set_systems_having_large_chromatic_number / problem_10
• erdos_1975_set_systems_having_large_chromatic_number / problem_9
• li_2026_resolution_erdos_problems_593_1177_obligatory
• li_2026_resolution_erdos_problems_593_1177_obligatory / corollary_1_3
• li_2026_resolution_erdos_problems_593_1177_obligatory / corollary_1_4
• li_2026_resolution_erdos_problems_593_1177_obligatory / theorem_1_1
• li_2026_resolution_erdos_problems_593_1177_obligatory / theorem_1_2