Coideals of block sequences

Abstract

We extend the well known notion of \textit{coideal} on $\mathbb{N}$ to families of block sequences on $FIN_k$ and prove that if a coideal of block sequences is \textit{semiselective} and satisfies a local version of Gowers' theorem \cite{Gow} then the local Ramsey property relative to it can be characterized in terms of the abstract Baire property, and the family of all sets having the local Ramsey property relative to one such coideal is closed under the Suslin operation. We also prove that these coideals satisfy a sort of \emph{canonical partition property} in the sense of Taylor \cite{taylor}, L\'opez-Abad \cite{jordi} and Blass \cite{blass}. This results give us an idea of the conditions to be considered in an abstract study of the local Ramsey property in the context of topological Ramsey spaces (see \cite{todo}).