Claims

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2026_01_05_barreto: Answers the site's wording (every family S meeting the hypothesis), not the corrected Statement (families closed under taking subgraphs), so it does not count toward the problem's standing. The remark, recorded in the site's commentary, is that a sparse family of complete graphs meets the hypothesis of Problem 638 while no graph has all its finite subgraphs in it.

2026_04_26_saturnino: Main Theorem 1 of a note posted to the site's thread on 26 April 2026: a hereditary class of finite graphs forcing monochromatic triangles under every finite number of colors but under no infinite cardinal; AI-assisted.

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