Abstract
Any regular mixed Tsirelson spaceT(θ n ,S n )N for whichθ n /θ n → 0, whereθ=lim n θ 1 , is shown to be arbitrarily distortable. Certain asymptoticl 1 constants for those and other mixed Tsirelson spaces are calculated. Also, a combinatorial result on the Schreier families (S α ) α<ω 1 is proved and an application is given to show that for every Banach spaceX with a basis (e i ), the two Δ-spectrums Δ(X) and Δ(X, (e i )) coincide.
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