  Let $(e_i)_i$ denote the unit vector basis of $\ell_p$, $1\leq p< \infty$, 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.
