= Bourgain l1-index
{c}
{title2=$I_1(X)=\sup_{K\in\mathbb N}o(T_K(X))$}
For $K\geq1$, let $T_K(X)$ consist of finite normalized sequences satisfying $K^{-1}\sum|a_i|\leq\|\sum a_ix_i\|\leq\sum|a_i|$ 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.
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