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We introduce the notion of (Ramsey) action of a tree on a (filtered)semigroup. We then prove in this setting a general result providing a commongeneralization of the infinitary Gowers Ramsey theorem for multiple tetrisoperations, the infinitary Hales--Jewett theorems (for both located andnonlocated words), and the Farah--Hindman--McLeod Ramsey theorem for layeredactions on partial semigroups. We also establish a polynomial version of ourmain result, recovering the polynomial Milliken--Taylor theorem ofBergelson--Hindman--Williams as a particular case. We present applications ofour Ramsey-theoretic results to the structure of delta sets in amenable groups.

Actions of trees on semigroups, and an infinitary Gowers--Hales--Jewett Ramsey theorem