On the complete separation of asymptotic structures in Banach spaces

Let $(e_i{)}_i$ denote the unit vector basis of ${\mathrm{\ell }}_p$, $1\le p<\mathrm{\infty }$, or $c_0$. We construct a reflexive Banach space with an unconditional basis that admits $(e_i{)}_i$ as a uniformly unique spreading model while it has no subspace with a unique asymptotic model, and hence it has no asymptotic-${\mathrm{\ell }}_p$ or $c_0$ subspace. This solves a problem of E. Odell. We also construct a space with a unique ${\mathrm{\ell }}_1$ spreading model and no subspace with a uniformly unique ${\mathrm{\ell }}_1$ spreading model. These results are achieved with the utilization of a new version of the method of saturation under constraints that uses sequences of functionals with increasing weights.