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

We study the nonlinear embeddability of Banach spaces and equi-embeddability family Kalton’s interlaced graphs [Formula: see text] into dual spaces. Notably, we define a modification property that call (with text]). show if equi-coarse Lipschitzly embeds text], then Szlenk index is greater than this optimal, i.e. there exists separable space contains equi-Lipschitzly so has text]. prove does not coarse embed by map with distortion strictly smaller also neither nor coarsely weak-to-weak[Formula: sequentially continuous map.