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

TL;DR

This paper provides a constructive proof of the finite Gowers' $FIN_k$ Theorem, analyzes its bounds, and compares it with the finite stabilization principle in finite-dimensional Banach spaces, especially $ ext{l}_ ext{infinity}^n$.

Contribution

It introduces a constructive proof of the finite Gowers' $FIN_k$ Theorem and compares its bounds with the finite stabilization principle in specific Banach spaces.

Findings

Constructive proof of finite Gowers' $FIN_k$ Theorem.

Analysis of upper bounds for the theorem.

Comparison showing slower growth of bounds in $ ext{l}_ ext{infinity}^n$ spaces.

Abstract

We give a constructive proof of the finite version of Gowers' $FIN_k$ Theorem and analyse the corresponding upper bounds. The $FIN_k$ 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 $FIN_k$ 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.