  We study density requirements on a given Banach space that guarantee the
existence of subsymmetric basic sequences by extending Tsirelson's well-known
space to larger index sets. We prove that for every cardinal $\kappa$ smaller
than the first Mahlo cardinal there is a reflexive Banach space of density
$\kappa$ without subsymmetric basic sequences. As for Tsirelson's space, our
construction is based on the existence of a rich collection of homogeneous
families on large index sets for which one can estimate the complexity on any
given infinite set. This is used to describe detailedly the asymptotic
structure of the spaces. The collections of families are of independent
interest and their existence is proved inductively. The fundamental stepping up
argument is the analysis of such collections of families on trees.
