Problem 602

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Statement. Let $(A_i)$ be a family of sets with $\lvert A_i\rvert=\aleph_0$ for all $i$, such that for any $i\neq j$ we have $\lvert A_i\cap A_j\rvert$ finite and $\neq 1$. Is there a $2$-colouring of $\cup A_i$ such that no $A_i$ is monochromatic?

Status. Open. The site's proof-claims tab carries one partial proof claim, filed 2026-10-01, a finite criterion for Property B that its author says does not resolve the problem; the site shows no verdict and labels the problem OPEN. The filing is recorded below and gets no claim page, since it settles no instance of the question.

Source. erdosproblems.com/602, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #602, https://www.erdosproblems.com/602.

Formalization. Statement in formal-conjectures.

Current assessment

The question (site formulation). The statement above; OPEN. The site's commentary attributes the problem to Komjáth and notes that the existence of such a $2$-coloring is called Property B. No assessment of the mathematics is recorded here beyond the known special cases below.

Known special cases. Countable families have Property B by Bernstein's lemma, which the formal-conjectures file states as a solved variant, so the question concerns uncountable families. Families whose pairwise intersections all have fewer than $n$ points, for a fixed finite $n$, have Property B by Miller's theorem (E. W. Miller, On a property of families of sets, C. R. Soc. Sci. Lett. Varsovie Cl. III 30 (1937), 31--38). Komjáth (Families close to disjoint ones, Acta Math. Hungar. 43 (1984), 199--207, doi:10.1007/BF01958019) strengthens this for such families of countable sets to essential disjointness: removing a finite set from each member leaves the members pairwise disjoint. Pairwise finite intersections alone do not suffice, since Miller's almost disjoint family on $\omega$ lacks Property B; so the hypothesis that no intersection has exactly one point does work. The case left open is that of uncountable families whose pairwise intersections are finite but not uniformly bounded. These classical results have no claim pages: none is credited by the site's commentary or the community database.

Claims. No claim page exists; the frontmatter standing is open with no claim. The one filing on the site's proof-claims page (claim 380, filed 2026-10-01 by Lezhe Gao under the forum name lezhe, with the systems DeepSeek and GPT-5.5 named in its entry) is the manuscript "Peeling Hypergraphs: A Structural Sufficient Condition for Property B", a Zenodo deposit of 2026-07-22 (doi:10.5281/zenodo.21475088). It calls a finite hypergraph peeling if its edges can be ordered so that each edge brings at least two vertices not in any earlier edge, or one such vertex together with a vertex of an earlier edge, proves that every peeling hypergraph has Property B, and characterizes peeling hypergraphs by acyclic assignments of vertices to edges; for graphs the peeling ones are exactly the forests. It gets no claim page because it settles no instance of the question, whose sets are countably infinite: the claimant's notes say that it is neither a proof of the infinite problem nor a partial solution to it, having no compactness argument, transfinite construction or reduction from the infinite case, and that it was filed as a related finite criterion in the hope that it prompts work on the infinite version. A result that decides no instance of the question is not a claim under the schema, whatever kind its filing carries.

Search scope, 2026-10-07: the site's problem page, commentary, discussion thread (one comment, post 5648 of 2026-04-20) and proof-claims page, with the Zenodo record of the claim. arXiv, Crossref, MathSciNet, zbMATH, Google Scholar and X were not searched.

Remaining gaps. (1) The mathematics of the question has not been assessed, and the literature on Property B for the stated families was not searched beyond the classical results named above. (2) The finite criterion filed on the site was not read beyond its record and the claim's summary.

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.

• erdos_1987_problems_finite_infinite_graphs
• erdos_1987_problems_finite_infinite_graphs / problem_12