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First, we prove a theorem on dynamics of actions of monoids by endomorphismsof semigroups. Second, we introduce algebraic structures suitable forformalizing infinitary Ramsey statements and prove a theorem that suchstatements are implied by existence of appropriate homomorphisms between thealgebraic structures. We make a connection between the two themes above, whichallows us to prove some general Ramsey theorems for sequences. We give a newproof of the Furstenberg--Katznelson Ramsey theorem; in fact, we obtain aversion of this theorem that is stronger than the original one. We answer inthe negative a question of Lupini on possible extensions of Gowers' Ramseytheorem.

Monoid actions and ultrafilter methods in Ramsey theory