Coalgebraic methods for Ramsey degrees of unary algebras.

In this paper we are interested in the existence of small and big Ramsey degrees of classes of finite unary algebras in arbitrary (not necessarily finite) algebraic language $\Omega$. We think of unary algebras as $M$-sets where $M = \Omega^*$ is the free monoid of words over the alphabet $\Omega$ and show that for an arbitrary monoid $M$ (finite or infinite) the class of all finite $M$-sets has finite small Ramsey degrees. This immediately implies that the class of all finite $G$-sets, where $G$ is an arbitrary group (finite or infinite), has finite small Ramsey degrees, and that the class of all finite unary algebras over an arbitrary (finite or infinite) algebraic language $\Omega$ has finite small Ramsey degrees. This generalizes some Ramsey-type results of M.\ Soki\'c concerning finite unary algebras over finite languages and finite $G$-sets for finite groups~$G$. To do so we develop a completely new strategy that relies on the fact that right adjoints preserve the Ramsey property. We then treat $M$-sets as Eilenberg-Moore coalgebras for ``half a comonad'' and using pre-adjunctions transport the Ramsey properties we are interested in from the category of finite or countably infinite chains of order type $\omega$. Moreover, we show that finite objects have finite big Ramsey degrees in the corresponding cofree structures over countably many generators.