  To any pair ( M , theta ) where M is a family of finite subsets of N compact
in the pointwise topology, and 0<theta < 1 , we associate a Tsirelson-type
Banach space T_M^theta . It is shown that if the Cantor-Bendixson index of M is
greater than n and theta >{1/n} then T_M^theta is reflexive. Moreover, if the
Cantor-Bendixson index of M is greater than omega then T_M^theta does not
contain any l^p, while if the Cantor-Bendixson index of M is finite
thenT_M^theta contains some l^p or c_o . In particular, if M ={ A subset N :
|A| leq n } and {1/n}<theta <1 then T_M^theta is isomorphic to some l^p .
