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Actions on semigroups and an infinitary Gowers–Hales–Jewett Ramsey theorem

We introduce the notion of (Ramsey) action on a (filtered) semigroup. We then prove in this setting a general result providing a common generalization of the infinitary Gowers Ramsey theorem for multiple tetris operations, the infinitary Hales–Jewett theorems (for both located and nonlocated words), and the Farah–Hindman–McLeod Ramsey theorem for layered actions on partial semigroups. We also establish a polynomial version of our main result, recovering the polynomial Milliken–Taylor theorem of Bergelson–Hindman–Williams as a particular case. We present applications of our Ramsey-theoretic results to the structure of recurrence sets in amenable groups.