# On the complete separation of asymptotic structures in Banach spaces

## TL;DR

This paper constructs specific Banach spaces demonstrating the separation between different asymptotic structures, solving a problem posed by Odell and revealing new insights into the geometry of Banach spaces.

## Contribution

It introduces a novel method of saturation under constraints to build Banach spaces with prescribed asymptotic properties, distinguishing between spreading models and asymptotic models.

## Key findings

- Constructed a reflexive Banach space with a unique spreading model but no asymptotic-$\, ext{l}_p$ or $c_0$ subspaces.
- Built a space with a unique $\, ext{l}_1$ spreading model but no uniformly unique $\, ext{l}_1$ spreading model.
- Developed a new version of the saturation method using sequences of functionals with increasing weights.

## Abstract

Let $(e_i)_i$ denote the unit vector basis of $\ell_p$, $1\leq p< \infty$, or $c_0$. We construct a reflexive Banach space with an unconditional basis that admits $(e_i)_i$ as a uniformly unique spreading model while it has no subspace with a unique asymptotic model, and hence it has no asymptotic-$\ell_p$ or $c_0$ subspace. This solves a problem of E. Odell. We also construct a space with a unique $\ell_1$ spreading model and no subspace with a uniformly unique $\ell_1$ spreading model. These results are achieved with the utilization of a new version of the method of saturation under constraints that uses sequences of functionals with increasing weights.