Families of Finite Subsets of ℕ of Low Complexity and Tsirelson Type Spaces

We study Tsirelson type spaces of the form T(ℳ︁_k, θ_k)^l_k=1] defined by a finite sequence (ℳ︁_k)^l_k=1 of compact families of finite subsets of ℕ. Using an appropriate index, denoted by i(ℳ︁), to measure the complexity of a family ℳ︁, we prove the following: If i(ℳ︁_k) < ω for all k = 1, …, l, then the space T(ℳ︁_k, θ_k)^l_k=1] contains isomorphically some l_p, 1 < p < ∞, or c₀. If i(ℳ︁) = ω, then the space Tℳ︁, θ] contains a subspace isomorphic to a subspace of the original Tsirelson's space.