On Matou\v{s}ek-like Embedding Obstructions of Countably Branching Graphs

TL;DR

This paper develops new metric-based proofs demonstrating the non-embeddability of countably branching trees and diamonds into specific Banach spaces, extending to curvature-like metric spaces and providing bounds on Lipschitz embeddings.

Contribution

It introduces entirely metric proofs for embedding obstructions of countably branching graphs into Banach spaces with property $(eta_p)$ and $p$-AMUC, extending results to curvature-like metric spaces.

Findings

Countably branching trees cannot embed into Banach spaces with property $(eta_p)$.

Countably branching diamonds cannot embed into $p$-AMUC Banach spaces for $p > 1$.

Lower bounds on Lipschitz embedding compression into spaces with $ ext{ell}_p$-asymptotic models.

Abstract

In this paper we present new proofs of the non-embeddability of countably branching trees into Banach spaces satisfying property $(\beta_p)$ and of countably branching diamonds into Banach spaces which are $p$-AMUC for $p > 1$. These proofs are entirely metric in nature and are inspired by previous work of Ji\v{r}\'i Matou\v{s}ek. In addition, using this metric method, we succeed in extending these results to metric spaces satisfying certain curvature-like inequalities. Finally, we extend an embedding result of Tessera to give lower bounds on the compression for a class of Lipschitz embeddings of the countably branching trees into Banach spaces containing $\ell_p$-asymptotic models for $p \geq 1$.