Families of finite subsets of IN of low complexity and Tsirelson type spaces

We study Tsirelson type spaces of the form T[(M-k,theta (k))(k=1)(l)] defin ed by a finite sequence (M-k)(k=1)(l) of compact families of finite subsets of IN. Using an appropriate index, denoted by i(M), to measure the complex ity of a family M, we prove the following: If i(M-k) < <omega> for all k = 1,...,l, then the space T[(M-k, theta (k))(k=1)(l)] contains isomorphically some l(p), 1 < p < infinity, or c(0). If I(M) = omega, then the space T[M, theta] contains a subspace isomorphic to a subspace of the original Tsirels on's space.