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Claim. There is a class $S_{\mathrm{ord}}$ of finite graphs, closed under isomorphism and under taking finite subgraphs, such that for every positive integer $n$ some member forces a monochromatic triangle under every coloring of its edges with $n$ colors, while for no infinite cardinal $\kappa$ is there a graph $G$, all of whose finite subgraphs lie in $S_{\mathrm{ord}}$, that forces a monochromatic triangle under every coloring with $\kappa$ colors; and there is a class $S_{\mathrm{ind}}$ with the same two properties for induced subgraphs. This is Main Theorem 1 of Saturnino's note, in the revised version posted to the site's thread on the evening of 26 April 2026, which the library's source card pages. The route, as the card records it: a graph forces a monochromatic triangle under $\kappa$ colors exactly when its triangle hypergraph has chromatic number above $\kappa$; the triangle case of the Nešetřil--Rödl sparse Ramsey theorem, the note's only external input, gives triangle-forcing graphs of large triangle-girth, which avoid any finite set of minimal two-color triangle-forcing graphs; blocks chosen recursively to avoid the minimal cores of the earlier blocks define the class; and a graph whose finite subgraphs all lie in the class has a triangle hypergraph of finite chromatic number, by the de Bruijn--Erdős compactness lemma or by finiteness.

Scope. Full: the class $S_{\mathrm{ord}}$ is closed under taking subgraphs and meets the hypothesis, so a correct proof answers no to the corrected Statement of Problem 638, which asks the question for such families; it is the only written argument on that Statement. The class $S_{\mathrm{ind}}$ answers the induced-subgraph variant, which the problem page records in its Formulation and does not count.

Standing. Claimed. The note is a forum posting: its author, under a pseudonymous account, posted it to the site's thread on 26 April 2026 after running it through Aristotle AI, an automated proof system; a commenter reported the same day that a standard check had found one minor mathematical issue and that the problem's intended interpretation might still be unclear; the author revised the note that evening, stating the hereditary scope in the abstract and fixing the length convention for Berge cycles, and on 27 April 2026 linked a Lean project that, in the author's words, formalizes the compactness and diagonal part of the ordinary-subgraph counterexample and leaves the finite avoidance principle and the block-sequence existence lemma as explicit sorrys, its initial draft produced with the same system. No refereed publication, arXiv version, independent mathematical review or complete formalization was found on 2026-09-18, the site's label and commentary do not mention the note, and its proof-claim tab is empty, so no evidence kind is listed. The source card checks the note's statements and not its proofs; nothing is independently reviewed. What would change the standing: an independent review or refereed publication of the note, a complete formalization, a second proof, or a counter-argument in the thread or the literature.

Postings. The site's discussion thread: the note's first version on 26 April 2026 (the first posting, which dates this page), the revised version the same evening, the version the source card cites, and the Lean project link on 27 April 2026. The Lean project is the GitHub repository linked above, pinned to its last commit of 27 April 2026 (its four commits all date from that day); the corpus has not built the Lean project. The note is hosted on a file-sharing site whose address embeds the poster's account name and is not printed here.