  We prove a local version of Gowers' Ramsey-type theorem [Ann. Math. 156
(2002)], as well as local versions both of the Banach space first dichotomy
(the "unconditional/HI" dichotomy) of Gowers [Ann. Math. 156 (2002)] and of the
third dichotomy (the "minimal/tight" dichotomy) due to Ferenczi-Rosendal [J.
Funct. Anal. 257 (2009)]. This means that we obtain versions of these
dichotomies restricted to certain families of subspaces called D-families, of
which several concrete examples are given. As a main example, non-Hilbertian
spaces form D-families; therefore versions of the above properties for
non-Hilbertian spaces appear in new Banach space dichotomies. As a consequence
we obtain new information on the number of subspaces of non-Hilbertian Banach
spaces, making some progress towards the "ergodic" conjecture of
Ferenczi-Rosendal and towards a question of Johnson.
