Coideals of block sequences

TL;DR

This paper generalizes the concept of coideals to block sequences on $FIN_k$, establishing their properties and connections to the local Ramsey property, Baire property, and canonical partition properties in topological Ramsey spaces.

Contribution

It introduces the notion of coideals of block sequences, proves their semiselectivity, and characterizes the local Ramsey property in this context, extending classical results.

Findings

Coideals of block sequences are semiselective.

The local Ramsey property is characterized by the abstract Baire property.

Coideals satisfy a canonical partition property.

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}).