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We introduce a notion of disjoint envelope functions to study asymptotic structures of Banach spaces. The main result gives a new characterization of asymptotic-${\ell }_p$ spaces in terms of the ${\ell }_p$-behavior of “disjoint-permissible” vectors of constant coefficients. Applying this result to Tirilman spaces we obtain a negative solution to a conjecture of Casazza and Shura. Further investigation of the disjoint envelopes leads to a finite-representability result in the spirit of the Maurey-Pisier theorem.

We generalize some results concerning the classical notion of a spreading model to spreading models of order ξ. Among other results, we prove that the set $S{M}_{\xi }^w\left(X\right)$ of ξ-order spreading models of a Banach space X generated by subordinated weakly null ℱ-sequences endowed with the pre-partial order of domination is a semilattice. Moreover, if $S{M}_{\xi }^w\left(X\right)$ contains an increasing sequence of length ω then it contains an increasing sequence of length ω₁. Finally, if $S{M}_{\xi }^w\left(X\right)$ is uncountable, then it contains an antichain of size...