Research notes on every problem, and a library of the papers behind them. Built from the open erdos repository.

Updated Oct 8

Problem 593

claims/: The 2 claim pages of Problem 593, one per claimant's result; the problem's standing derives from them.

Statement. Characterize those finite 3-uniform hypergraphs which appear in every 3-uniform hypergraph of chromatic number $>\aleph_0$ .

Status. OPEN, the site's label. The site's proof-claims tab carries two full proof claims, both found with AI systems and both with Lean developments; the site shows no verdict on either and its label is unchanged (proof-claims thread accessed 2026-10-06). The derived standing departs from the label because the two claims are pending full claims that give the same characterization, recorded on their claim pages, so the problem is claimed as answered; neither is refereed or reviewed by anyone the record names, so neither is accepted.

Source. erdosproblems.com/593, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #593, https://www.erdosproblems.com/593.

References.

[EGH75] Erdős, P. and Galvin, F. and Hajnal, A., On set-systems having large chromatic number and not containing prescribed subsystems. (1975), 425-513.

Formalization. Statement in formal-conjectures.

Current assessment

The question (site formulation). The statement above; OPEN. The site's commentary points to Erdős, Galvin and Hajnal [EGH75] for related problems and records Erdős's view that the graph case is settled: a graph of chromatic number at least $\aleph_1$ contains every finite bipartite graph but need not contain any given odd cycle. The question asks for the analogous characterization for finite $3$ -uniform hypergraphs.

Claims. Two pending full claims give the same answer: a finite triple system is forced by uncountable chromatic number exactly when, isolated vertices removed, it is linear, every edge-node of its Levi graph lies on a bridge and every Berge cycle is even. Li's classification (arXiv preprint of 2026-06-23, filed on the site on 2026-07-17) and Petkov's alternative proof (filed 2026-07-15, manuscript distributed from the claimant's repository, which credits Li's preprint as the first complete proof) were both found with AI systems, named on the claim pages, and both come with Lean developments that this corpus has not built. Neither is refereed or reviewed by anyone the record names, and the site shows no verdict, so both stay claimed; the two pending claims agree, which is what the frontmatter records. Li's paper also claims the answers to Problem 1177; that claim belongs on that problem's page.

Search scope, 2026-10-07: the site's problem page, commentary, discussion thread and proof-claims page, with the comments the site lists on the two claims (three and seven); the arXiv record of Li's preprint; the READMEs of both repositories and their commit records for dates and pins. arXiv beyond the record, Crossref, MathSciNet, zbMATH, Google Scholar and X were not searched.

Remaining gaps. (1) Neither manuscript has a referee or a named reviewer. (2) Neither Lean development was built or audited here, so neither claim page lists formalized evidence, and whether Petkov's Lean definitions reach the uncountable hosts of the question was not examined. (3) The two proofs are not informationally independent by the second claimant's later account: in a site comment of 2026-07-15 Petkov called his proof independent, but his manuscript revision of 2026-07-21 presents an alternative proof that reuses Li's one-apex sequence-lift and bridge-trace framework for the converse, credits Li's preprint as the first complete proof and claims no informational independence.

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.

Linked from (15)

Li's classification of the obligatory finite triple systems Petkov's alternative proof of the triple system classification problems/set_theory/E0593/claims Problem 1177 Li's exact spectra of triple systems avoiding a fixed one Set Theory and Infinite Combinatorics graph_coloring/erdos_1995_problems_combinatorial_set_theory Section 6 (p. 64): triple systems of uncountable chromatic number and the finite triple systems they must contain Set Theory and Infinite Combinatorics set_theory/erdos_1975_set_systems_having_large_chromatic_number Problem 10 (p. 498): the finite triple systems that occur in every triple system of chromatic number above aleph_0 Theorem 14.4 (p. 495): the function g-check_3 bounds the densest t-point subsystems forced by uncountable chromatic number Theorem 3.8 (p. 438) and Corollary 3.9: two unavoidable configurations in triple systems of large chromatic number set_theory/li_2026_resolution_erdos_problems_593_1177_obligatory Theorem 1.1 (p. 1): a finite triple system is obligatory iff it lies in the class B iff it is linear with a bridge at every edge-node and only even Berge cycles

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