  We discuss pruning and coloring lemmas on regular families. We discuss
several applications of these lemmas to computing the Szlenk index of certain
$w^*$ compact subsets of the dual of a separable Banach space. Applications
include estimates of the Szlenk index of Minkowski sums, infinite direct sums
of separable Banach spaces, constant reduction, and three space properties.
  We also consider using regular families to construct Banach spaces with
prescribed Szlenk index. As a consequence, we give a characterization of which
countable ordinals occur as the Szlenk index of a Banach space, prove the
optimality of a previous universality result, and compute the Szlenk index of
the injective tensor product of separable Banach spaces.
