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Claim. The preprint's abstract describes characteristic earlier results as consistency statements $\mathrm{Con}(2^{\aleph_0}\to[\lambda]^2_{n,2})$ in which the continuum is a former large cardinal, at best the first weakly Mahlo cardinal, and states the new results as $\mathrm{Con}\big((2^{\aleph_0}=\aleph_m)+\aleph_l\to[\aleph_k]^2_{n,2}\big)$ for suitable $k<l<m$: the continuum may be small, no large cardinal is used, and the cardinals carrying the relation lie below the continuum after the forcing. A comment in the discussion thread of Problem 474 (2026-08-17) reads the paper as resolving the problem as independent of ZFC without any large cardinal assumption; the comment says that the AI system Sol extracted the relevant claim from the paper's summary of results. Read that way, the paper removes the large cardinal hypothesis of the accepted claim Shelah 1988: a $3$-coloring of the pairs of the plane in which every uncountable set realizes all three colors is not provable to exist in ZFC, relative to the consistency of ZFC alone. Under the continuum hypothesis the coloring exists, by Erdős's construction recorded on Erdős, Hajnal and Rado 1965, so the existence of the coloring is independent of ZFC, which is the claim value of this page.

What the abstract settles. Positive square-bracket relations transfer upward in the first cardinal and downward in the second and in the number of colors: if $\aleph_l\to[\aleph_k]^2_{n,2}$ holds with $\aleph_l\le2^{\aleph_0}$, $k\ge1$ and $n\ge3$, then every $3$-coloring of the pairs of a set of size continuum restricts to one of a subset of size $\aleph_l$, which has a subset of size $\aleph_1$ realizing at most two colors, so $2^{\aleph_0}\to[\aleph_1]^2_3$. Since the abstract places the cardinals $\aleph_l$ below the continuum, the only points the abstract leaves unchecked are whether the paper's $k$ and $n$ include $k\ge1$ and $n\ge3$. The condition $k<l<m$ with $k\ge1$ forces $2^{\aleph_0}=\aleph_m\ge\aleph_3$, so the preprint does not touch the problem page's question with $2^{\aleph_0}=\aleph_2$. No acceptance evidence is recorded: the curator's problem page, last edited 1 February 2026, predates the thread comment and does not cite the preprint, and no refereed version is recorded.

Source. Saharon Shelah, Consistency of square bracket partition relation, arXiv:2601.02923, version 1 submitted 2026-01-06, 14 pages.

Depends on. The independence value uses the CH direction recorded on Erdős, Hajnal and Rado 1965; the unprovability half is the claim of Shelah 1988 with its large cardinal removed.