 Gowers dichotomy theorem

Every infinite-dimensional Banach space contains either an unconditional basic sequence or a hereditarily indecomposable Banach space. For a space with a basis and no unconditional basic sequence, apply the block Ramsey theorem to finite sequences whose odd and even spans contain nearby unit vectors. A diagonal sequence of successively sharper games gives a subspace in which every pair of infinite-dimensional subspaces has angle zero.