Proximity to $\ell_p$ and $c_0$ in Banach spaces

TL;DR

This paper introduces new minimal trees and coloring theorems to analyze the proximity of Banach spaces to classical sequence spaces like _p and c_0, providing tools for quantifying and estimating their indices.

Contribution

It develops a novel class of minimal trees and coloring theorems to improve the understanding and quantification of Bourgain _p and c_0 indices in Banach spaces.

Findings

Quantification of Bourgain _p and c_0 indices

Dualization of Bourgain c_0 index

Estimates of _p index via subspaces and quotients

Abstract

We construct a class of minimal trees and use these trees to establish a number of coloring theorems on general trees. Among the applications of these trees and coloring theorems are quantification of the Bourgain $\ell_p$ and $c_0$ indices, dualization of the Bourgain $c_0$ index, establishing sharp positive and negative results for constant reduction, and estimating the Bourgain $\ell_p$ index of an arbitrary Banach space $X$ in terms of a subspace $Y$ and the quotient $X/Y$.