On Kalton's interlaced graphs and nonlinear embeddings into dual Banach spaces

We study the nonlinear embeddability of Banach spaces and the equi-embeddability of the family of Kalton’s interlaced graphs $([\mathbb N]^k,d_{\mathbb K})k$ into dual spaces. Notably, we define and study a modification of Kalton’s property $\mathcal{Q}$ that we call property ${\mathcal{Q}}_p$ (with $p\in (1,+\mathrm{\infty }]$). We show that if $([\mathbb N]^k,d{\mathbb K})k$equi-coarseLipschitzlyembedsinto$X^*$, then the Szlenk index of $X$ is greater than $\omega$, and that this is optimal, i.e., there exists a separable dual space $Y^{\ast }$ that contains $([\mathbb N]^k,d{\mathbb K})_k$equi-Lipschitzlyandsothat$Y$hasSzlenkindex$\omega^2$\mathrm{.}Weprovethat$c_0$ does not coarse Lipschitzly embed into a separable dual space by a map with distortion strictly smaller than $\frac{3}{2}$. We also show that neither $c_0$ nor $L_1$ coarsely embeds into a separable dual by a weak-to-weak${}^{\ast }$ sequentially continuous map.