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Claim. Problem 638 asks whether every family $S$ of finite graphs that contains, for each $n$, a graph forcing a monochromatic triangle under every coloring of its edges with $n$ colors must also contain, for every infinite cardinal $\aleph$, all the finite subgraphs of some graph $G$ forcing a monochromatic triangle under every coloring with $\aleph$ colors. Barreto's remark, written into the site's commentary by its curator, is that this is false as worded unless $S$ is also closed under taking subgraphs: let $S$ consist of one complete graph for each $n$, large enough that every $n$-coloring of its edges has a monochromatic triangle (the thread names the complete graphs of the successive triangle Ramsey numbers). The hypothesis holds. But a graph $G$ with two or more vertices has a subgraph on two vertices with no edge, which is not complete and so not in $S$, and a graph on one vertex contains no triangle; so for no infinite cardinal does the graph asked for exist, and the answer to the site's wording is no. The argument uses nothing about triangles or colorings beyond the hypothesis: it rests on $S$ not being closed under subgraphs. (The same family also misses complete graphs of most orders, so the variant with induced subgraphs fails for it too; an authored remark that warrants nothing here.)

Why it is rejected. It answers the site's wording, not the corrected statement. Problem 638 judges the corrected Statement, in which $S$ is closed under taking subgraphs; the problem page's Notes give the evidence. The sparse family of complete graphs is not closed under subgraphs, so the counterexample settles no instance of the corrected Statement. The problem page's Notes credit the result.

Depends on. Nothing in this wiki; the argument is the site's own observation.

Acceptance. None recorded. The site's curator, T. F. Bloom, records the remark in the problem's commentary, credits Barreto by name, and tags the thread comment as addressed by an update of the page (page last edited 10 April 2026), but the site's label stays OPEN, which fits the corrected Statement, and commentary on a problem so labeled is not an acceptance under the corpus's rule; no refereed publication or formalization states the counterexample. The observation is elementary. The thread's later comments (22 April to 27 April 2026) take the failure of the site's wording as given and turn to the hereditary reading; none disputes the remark, and the proof-claim tab is empty.

Postings and dating. The thread comment of 5 January 2026 is the first posting and names this page; the commentary carries no date of its own. The site keeps the problem labeled OPEN for the hereditary reading.