Actions of trees 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.