A new hierarchy of Banach spaces T k (d, θ), k any positive integer, is constructed using barriers in high dimensional Ellentuck spaces [9] following the classical framework under which a Tsirelson type norm is defined from a barrier in the Ellentuck space [3]. The following structural properties of these spaces are proved. Each of these spaces contains arbitrarily large copies of n ∞ , with the bound constant for all n. For each fixed pair d and θ, the spaces T k (d, θ), k ≥ 1, are p-saturated, forming natural extensions of the p space, where p satisfies dθ = d 1/p . Moreover, they form a strict hierarchy over the p space: For any j < k, the space T j (d, θ) embeds isometrically into T k (d, θ) as a subspace which is non-isomorphic to T k (d, θ).

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