math.LO — Logic
Ramsey’s Theorem for Pairs in Pincus’s Model XI
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 transfer theorems, yields Con(ZF) ⇒ Con(ZF + RT_pairs + DO + BPI + ¬Part∞), where Part∞ states that every infinite set is the disjoint union of infinitely many infinite sets.