This is an announcement for the paper "On the hereditary proximity to $\ell_1$" by Spiros A. Argyros, A. Manoussakis, and Anna M. Pelczar. Abstract: 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. Archive classification: math.FA Mathematics Subject Classification: 46B20; 46B15; 03E10; 05A17 Remarks: 40 pages, submitted for publication The source file(s), proximity.tex: 158273 bytes, is(are) stored in gzipped form as 0907.4317.gz with size 43kb. The corresponding postcript file has gzipped size 238kb. Submitted from: anna.pelczar@im.uj.edu.pl The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/0907.4317 or http://arXiv.org/abs/0907.4317 or by email in unzipped form by transmitting an empty message with subject line uget 0907.4317 or in gzipped form by using subject line get 0907.4317 to: math@arXiv.org.