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 1 (respectively,c0) thenXcontains 1(respectively,c0). Furthermore, Tsirelson's 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 1. An equivalent norm is constructed onTso that s1+s2<2 whenever (sn) is a spreading model of a normalized basic sequence inT. 1998 Academic Press.