Finite forms of Gowers' Theorem on the oscillation stability of $c_0$

Abstract

We give a constructive proof of the finite version of Gowers' $FI{N}_k$ Theorem and analyse the corresponding upper bounds. The $FI{N}_k$ Theorem is closely related to the oscillation stability of $c_0$. The stabilization of Lipschitz functions on arbitrary finite dimensional Banach spaces was studied well before by V. Milman. We compare the finite $FI{N}_k$ Theorem with the finite stabilization principle in the case of spaces of the form ${\ell }_{\infty }^n$, $n\in \mathbb{N}$ and establish a much slower growing upper bound for the finite stabilization principle in this particular case.

arXiv comment18 pages

Cited by in corpus (1)

• Combinatorial proofs of Gowers' FIN${}_k$ and FIN${}_k^{\pm }$ theorem