  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 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 delta sets in amenable groups.
