A Problem on Spreading Models

It is proved that if a Banach spaceXhas a basis (en) satisfying every spreading model of a normalized block basis of (en) is 1-equivalent to the unit vector basis of ℓ1 (respectively,c0) thenXcontains ℓ1(respectively,c0). Furthermore, Tsirelsonʹs spaceTis shown to have the property that every infinite dimensional subspace contains a sequence having spreading model 1-equivalent to the unit vector basis of ℓ1. An equivalent norm is constructed onTso that ‖s1+s2‖<2 whenever (sn) is a spreading model of a normalized basic sequence inT.