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 FIN ±k .

Parametrized Ramsey theory of infinite block sequences of vectors

“…Similarly, the following result from [9] gives an infinite-dimensional version of Gowers' FIN ±k theorem.…”

“…±k theorem In this section we prove the following approximate Ramsey theorem, which parametrizes the infinite-dimensional version of Gowers' FIN ±k theorem from [9]. First, given two infinite block sequences A = (a n ) and…”

“…However, our approach has the advantage of isolating useful ultrafilters which may be of independent interest and which, in some sense, provide an explanation for the existence of the corresponding Ramsey results without relying on the companion result for FIN k . Our hope is that the methods used in this paper (and the related paper [9], both of which are in turn based on [17]) will eventually lead to a unified approach to obtaining approximate Ramsey results, even in cases where there is no naturally associated "exact" result or when there is no way to transfer the exact results to the approximate setting.…”