Banach spaces from barriers in high dimensional Ellentuck spaces

A new hierarchy of Banach spaces $T_k(d,theta)$, $k$ any positive integer, is constructed using barriers in high dimensional Ellentuck spaces cite{DobrinenJSL15} following the classical framework under which a Tsirelson type norm is defined from a barrier in the Ellentuck space cite{Argyros/TodorcevicBK}.  The following structural properties of these spaces are proved.  Each of these spaces contains arbitrarily large copies of $ell_infty^n$, with the bound constant for all $n$.  For each fixed pair $d$ and $theta$, the spaces $T_k(d,theta)$, $kge 1$, are $ell_p$-saturated, forming natural extensions of the $ell_p$ space, where $p$ satisfies $dtheta=d^{1/p}$.  Moreover, they form a strict hierarchy over the $ell_p$ space: For any $ju003ck$, the space $T_j(d,theta)$ embeds isometrically into $T_k(d,theta)$ as a subspace which is non-isomorphic to $T_k(d,theta)$.