Minimality, homogeneity and topological 0-1 laws for subspaces of a Banach space

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If a Banach space is saturated with basic sequences whose linear span embeds into the linear span of any subsequence, then it contains a minimal subspace. It follows that any Banach space is either ergodic or contains a minimal subspace. For a Banach space X with an (unconditional) basis, topological 0-1 law type dichotomies are stated for block-subspaces of X as well as for subspaces of X with a successive FDD on its basis. A uniformity principle for properties of block-sequences, results about block-homogeneity, and a possible method to construct a Banach space with an unconditional basis, which has a complemented subspace without an unconditional basis, are deduced.