  In the first part of the paper we present and discuss concepts of local and
asymptotic hereditary proximity to \ell_1. The second part is devoted to a
complete separation of the hereditary local proximity to \ell_1 from the
asymptotic one. More precisely for every countable ordinal \xi we construct a
separable reflexive space \mathfrak{X}_\xi such that every infinite dimensional
subspace of it has Bourgain \ell_1-index greater than \omega^\xi and the space
itself has no \ell_1-spreading model. We also present a reflexive HI space
admitting no \ell_p as a spreading model.
