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Abstract: A well-known theorem of Mathias says that no infinite maximal almost disjoint, or "mad", family of infinite subsets of the natural numbers can be analytic. Mathias' proof utilizes a relationship between mad families and the "local" Ramsey theory of the natural numbers. We consider the analogous question for maximal almost disjoint families of infinite-dimensional subspaces of a countable, infinite-dimensional vector space, and connect it to the Ramsey theory of block sequences in such spaces, first developed by Gowers for Banach spaces.