  We extend the methods used by V. Ferenczi and Ch. Rosendal to obtain the
`third dichotomy' in the program of classification of Banach spaces up to
subspaces, in order to prove that a Banach space E with an admissible system of
blocks with admissible set A, contains an infinite dimensional subspace with a
basis which is either A-tight or A-minimal. In this setting we obtain, in
particular, dichotomies regarding subsequences of a basis, and as a corollary,
we show that every normalized basic sequence has a subsequence which either
satisfies a tightness property or is spreading. Other dichotomies between
notions of minimality and tightness are demonstrated, and the Ferenczi-Godefroy
interpretation of tightness in terms of Baire category is extended to this new
context.
