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

## TL;DR

This paper extends Gowers' finite-dimensional Ramsey theorem to an infinite-dimensional setting using ultra-Ramsey spaces, with applications to the stability of Lipschitz functions on the unit sphere of c0.

## Contribution

It introduces an infinite-dimensional version of Gowers' theorem employing ultra-Ramsey space theory, expanding the scope of combinatorial and functional analysis results.

## Key findings

- Established an Ellentuck-type theorem for infinite block sequences in FIN_{±k}
- Demonstrated the theorem's application to Lipschitz function stability on c0
- Extended finite-dimensional Ramsey theory to an infinite-dimensional context

## 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 $\mathrm{FIN}_{\pm k}$.