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The formulation provides an elementary axiomatics for the methods of abstract ultra-Ramsey theory.\n  The main results are Theorem 10, Theorem 57, Theorem 67 and Theorem 73. Theorem 10 is an infinite-dimensional extension of the celebrated Ramsey's Theorem. We show that corollaries of this result include the Galvin-Pirky Theorem, the Silver Theorem and the $\\vec{\\alpha}$-Ellentuck Theorem. 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