An infinite-dimensional version of Gowers' ${\mathrm{FIN}}_{\pm k}$ theorem

Abstract

We prove an infinite-dimensional version of an approximate Ramsey theorem of Gowers, initially used to show that every Lipschitz function on the unit sphere of $c_0$ is oscillation stable. To do so, we use the theory of ultra-Ramsey spaces developed by Todorcevic in order to obtain an Ellentuck-type theorem for the space of all infinite block sequences in ${\mathrm{FIN}}_{\pm k}$.

15 pages; fixed typos, clarified notation and added references