{"id":"hexagon/2610.00464","pdf":"https://hexagonmath.org/pdf/2610.00464v2","url":"https://hexagonmath.org/abs/2610.00464v2","tags":["msc:03E25","msc:03E35","msc:03E02","msc:05D10","axiom of choice; Ramsey's theorem"],"score":0,"title":"Ramsey’s Theorem for Pairs in Pincus’s Model XI","domain":"math.LO","origin":"ai-generated","source":"hexagon","authors":["Frank Gilson"],"reviews":0,"subject":"math.LO","summary":"Howard and Rubin record Pincus’s ordered colored permutation model as N48, but do not determine whether Ramsey’s theorem for pairs holds there. We prove that N48 |= RT for pairs. The proof uses least finite supports and the homogeneity of Pincus’s ordered colored structure: after fixing all but one point of a support, the remaining point generates an infinite orbit whose unordered pairs form a single stabilizer orbit. Combining this with Pincus’s results on the dense linear ordering principle (DO), the Boolean Prime Ideal Theorem (BPI), and the failure of Part∞, together with Pincus’s…","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"ai_models":["ChatGPT","Claude Opus","meta.ai","apodex.ai Deep Solve"],"discussed":0,"published":"2026-10-22","reactions":{},"withdrawn":false,"authorship":"mixed","formalized":false,"human_understanding":"some"}