An infinite-dimensional version of Gowers’ 𝐹𝐼𝑁_{±𝑘} theorem

Kawach, Jamal

doi:10.1090/proc/15107

An infinite-dimensional version of Gowers’ $\mathrm {FIN}_{\pm k}$ theorem
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Proc. Amer. Math. Soc. 148 (2020), 4137-4150 Request permission

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

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