Bourgain l1-index ($I_1(X)={\sup }_{K\in \mathbb{N}}o(T_K(X))$)

For $K\ge 1$, let $T_K(X)$ consist of finite normalized sequences satisfying $K^{-1}\sum |a_i|\le \Vert \sum a_i{x}_i\Vert \le \sum |a_i|$ for every scalar combination. Its tree order measures the depth of finite copies of the unit vector basis of ${\ell }^1$. A separable Banach space containing no copy of ${\ell }^1$ has a countable index because every such tree is closed and well-founded. Countable well-founded trees can be used to construct separable reflexive spaces with arbitrarily high index, precluding a separable reflexive universal space.