On the complete separation of asymptotic structures in Banach spaces

Let $(e_i)_i$ denote the unit vector basis of $ell_p$, $1leq p< infty$, 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-$ell_p$ or $c_0$ subspace. This solves a problem of E. Odell. We also construct a space with a unique $ell_1$ spreading model and no subspace with a uniformly unique $ell_1$ 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.