On Matoušek-like embedding obstructions of countably branching graphs

In this paper we present new proofs of the non-embeddability of countably branching trees into Banach spaces satisfying property $\left({\beta }_p\right)$ and of countably branching diamonds into Banach spaces which are ${\ell }_p$-asymptotic midpoint uniformly convex (p-AMUC) for $p>1$. These proofs are entirely metric in nature and are inspired by previous work of Jiří Matoušek. In addition, using this metric method, we succeed in extending these results to metric spaces satisfying certain embedding obstruction inequalities. Finally, we give Tessera-type 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 $p\ge 1$.