On antichains of spreading models of Banach spaces

TL;DR

This paper proves that for any separable Banach space, the set of spreading models generated by weakly-null sequences is either countable or contains a large antichain, answering a question in Banach space theory.

Contribution

It establishes a dichotomy for the structure of spreading models in separable Banach spaces, showing the set is either countable or contains a continuum-sized antichain.

Findings

The set of spreading models is either countable or contains a continuum-sized antichain.

Answers an open question by Dilworth, Odell, and Sari.

Provides a structural insight into the complexity of spreading models.

Abstract

We show that for every separable Banach space $X$, either $\spw(X)$ (the set of all spreading models of $X$ generated by weakly-null sequences in $X$, modulo equivalence) is countable, or $\spw(X)$ contains an antichain of the size of the continuum. This answers a question of S. J. Dilworth, E. Odell and B. Sari.