On asymptotic models in Banach spaces

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A well known application of Ramsey's Theorem to Banach Space Theory is the notion of a spreading model (e'ᵢ) of a normalized basic sequence (xᵢ) in a Banach space X. We show how to generalize the construction to define a new creature (eᵢ), which we call an asymptotic model of X. Every spreading model of X is an asymptotic model of X and in most settings, such as if X is reflexive, every normalized block basis of an asymptotic model is itself an asymptotic model. We also show how to use the Hindman-Milliken Theorem--a strengthened form of Ramsey's Theorem--to generate asymptotic models with a stronger form of convergence.