A classification of Tsirelson type spaces

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We give a complete classification of mixed Tsirelson spaces T[(F_i, theta_i)_i=1ʳ] for finitely many pairs of given compact and hereditary families F_i of finite sets of integers and 0<theta_i<1 in terms of the Cantor-Bendixson indexes of the families F_i, and theta_i (0< i < r+1). We prove that there are unique countable ordinal alpha and 0<theta<1 such that every block sequence of T[(F_i, theta_i)_i=1ʳ] has a subsequence equivalent to a subsequence of the natural basis of the T(S_omegaᵃlpha,theta). Finally, we give a complete criterion of comparison in between two of these mixed Tsirelson spaces.