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Claim. Write $\kappa=(2^{\aleph_0})^+$ and read $[\lambda]^\kappa$ as the subsets of size $\kappa$, the site's reading. The answer posted on MathOverflow (question 511508, "On Erdős Problem #598") argues that $\lambda\to[\kappa]^\omega_\kappa$ for an uncountable $\lambda\ge\kappa$ implies that $0^\sharp$ exists: the relation gives a weaker relation on finite subsets, equivalent to the existence, for every structure on a set $M$ of size $\lambda$ with a distinguished $P\subseteq M$ of size $\kappa$, of an elementary substructure $N$ of size $\kappa$ with $N\cap P$ a proper subset of $P$; applied to $L_\lambda$ with $P=\kappa$, the transitive collapse of $N$ gives a nontrivial elementary embedding $L_\alpha\to L_\lambda$ with critical point below $\kappa$, and such an embedding yields $0^\sharp$ (Jech, Set Theory, Theorem 18.27). Hence in every model without $0^\sharp$, in particular in $L$, the coloring asked by Problem 598 exists for every infinite $m$: the answer is yes. On the other side, the forcing of Garti and Hayut (The first omitting cardinal for Magidority, Math. Log. Q. 65 (2019), 95–104; library card) gives, from the consistency of an elementary embedding $j:V_{\lambda+1}\to V_{\lambda+1}$ (the axiom I1), a model with $\kappa=\aleph_2$ in which $\lambda\to[\kappa]^\omega_\kappa$ for some $\lambda$, a negative instance. So the problem, read as one question about every infinite $m$, is independent of ZFC relative to the consistency of I1.

Submission note. Posted to the site's forum by Fanxin Wu on 7 June 2026:

Based on this MO discussion the problem is independent modulo large cardinals. More precisely, the positive answer holds in the constructible universe $L$ so is consistent. The negative answer is consistent assuming the consistency of something called rank-into-rank axiom, which is pretty high in the large cardinal hierarchy; the exact large cardinal needed is probably open, but it's at least zero sharp.

I would also like to propose distinguishing problems like this from problems that are independent but don't involve large cardinals, namely #1119 and #1127.

Hypothesis. The negative half is relative to a rank-into-rank axiom, far above the consistency strength of ZFC; the positive half holds in $L$ outright, since $0^\sharp$ does not exist there. The answer leaves open the exact consistency strength of $\lambda\to[\kappa]^\omega_\kappa$, which lies between $0^\sharp$ and I1, and notes that Laver's earlier consistency result may use the other reading of $[\lambda]^\kappa$, by order type. Wu calls the independence essentially folklore.

Source. Fanxin Wu, answer of 2026-05-24 to Wu's own MathOverflow question 511508; the answer is marked accepted by the asker, the same person, so the mark is not review. The author summarized it on the site's discussion thread on 2026-06-07 (post 6877). Not refereed. Nothing on this page is independently reviewed by this project.

Standing. No reviewer is recorded and the site labels the problem OPEN, so the claim stays claimed. Chojecki's note of 2026-04-22 obtained the negative model earlier from the same Garti–Hayut forcing, without the half in $L$, and White's report of 2026-07-28 reproves both halves and cites this answer.