Problem 598

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claims/: The 3 claim pages of Problem 598, one per claimant's result; the problem's standing derives from them.

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Statement. Let $m$ be an infinite cardinal and $\kappa$ be the successor cardinal of $2^{\aleph_0}$. Can one colour the countable subsets of $m$ using $\kappa$ many colours so that every $X\subseteq m$ with $\lvert X\rvert=\kappa$ contains subsets of all possible colours?

Status. Open. The site labels the problem OPEN, with no commentary and no proof claim; the results posted on its discussion thread and outside the site are recorded in the Current assessment and on the claim pages.

Source. erdosproblems.com/598, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #598, https://www.erdosproblems.com/598.

Formalization. Statement in formal-conjectures.

Current assessment

The question (site formulation). The statement above, labeled OPEN on the site, with no commentary and no proof claim. Erdős's own wording, in Problem 8 of Erdős 1987 (printed pp. 225–226), asks whether for every infinite $m$ one can color the countable subsets of $m$ by $(2^{\aleph_0})^+$ colors so that every subset of size $(2^{\aleph_0})^+$ gets subsets of all the colors. The site fixes $m$ and asks the question for it, so a model with one failing $m$ is a negative instance of the site's question and a negative answer to Erdős's; the claim pages read the problem Erdős's way, as one question about every infinite $m$, and the instances are recorded separately here.

Formulation. In the notation of the claim pages, write $\kappa=(2^{\aleph_0})^+$ and $\mathrm{Col}_\omega(m,\kappa)$ for the property that some coloring $c:[m]^\omega\to\kappa$ gives every $X\in[m]^\kappa$ countable subsets of all $\kappa$ colors; in square-bracket notation this is $m\not\to[\kappa]^\omega_\kappa$, with $[m]^\kappa$ the subsets of size $\kappa$.

In ZFC. The property holds for every $m\le\kappa$: vacuously for $m<\kappa$, which has no subset of size $\kappa$, and for $m=\kappa$ by a coloring built from Solovay's partition of the ordinals of countable cofinality below $\kappa$ into $\kappa$ stationary sets (Proposition 2.1 of Chojecki's note, the repaired form of thread post 4782 of 2026-03-15 by Zeraoulia Rafik, which a reply of the same day, post 4801, reported to follow from work of Garti and Hayut). Rafik's post gets no claim page: it is a thread post, not a dated manuscript, and the note carries the result. The note's Corollary 2.8 states the range as $m\le\kappa^\omega$, which equals $\kappa$ by Hausdorff's formula, so it adds no instance.

Relative independence. Read as one question about every infinite $m$, the problem is independent of ZFC relative to large cardinals. Wu's answer (2026-05-24) shows that a failure for any $\lambda$ implies that $0^\sharp$ exists, so the answer is yes for every $m$ in $L$, and that the forcing of Garti and Hayut from the rank-into-rank axiom I1 gives a model with a failing $m$. Chojecki's note (2026-04-22, written with GPT-5.4 Pro) obtained the negative model first from the same forcing, under unspecified large-cardinal hypotheses, with a threshold cardinal $\lambda^*$ from which on every $m$ fails; its companion Lean file formalizes the positive constructions and states the threshold theorem as True. White's report (2026-07-28, written with Claude (Anthropic), labeled PARTIAL by its ledger) reproves both halves and shows that any least failing cardinal is regular and $\aleph_0$-closed. The exact consistency strength of a failure lies between $0^\sharp$ and I1 and is not determined.

Claims. Three pending conditional claims, the pages named above; none is reviewed or refereed. A conditional claim derives no standing, so the frontmatter standing is open with no claim.

Search scope. The site's problem page, its discussion thread (eight comments) and proof-claims tab (none), MathOverflow question 511508, the erdosproblemaday ledger and the formal-conjectures statement file were read on 2026-10-07. arXiv, Crossref, MathSciNet, zbMATH and Google Scholar were not searched.

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_8
• garti_2019_first_omitting_cardinal_magidority
• garti_2019_first_omitting_cardinal_magidority / claim_1_10