The following property of a normalized basis in a Banach space is considered: any normalized block sequence of the basis has a subsequence equivalent to the basis. Under uniformity or other natural assumptions, a basis with this property is equivalent to the unit vector basis of c 0 or p . An analogous problem concerning spreading models is also addressed.

Abstract. We show that every Banach space saturated with subsymmetric basic sequences contains a minimal subspace.

Key words Mixed Tsirelson space, p-space, quasiminimality MSC (2010) 46B20, 46B15We prove quasiminimality of regular the mixed Tsirelson spaces T [(Sn , θn )n ] with the sequence θ n θ n n decreasing, where θ = limn θ 1 / n n , and quasiminimality of all mixed Tsirelson spaces T [(An , θn )n ]. We prove that under certain assumptions on the sequence (θn )n the dual spaces are quasiminimal.