On the diameter of the Banach-Mazur set

On every subspace of l _∞(ℕ) which contains an uncountable ω-independent set, we construct equivalent norms whose Banach-Mazur distance is as large as required. Under Martin’s Maximum Axiom (MM), it follows that the Banach-Mazur diameter of the set of equivalent norms on every infinite-dimensional subspace of l _∞(ℕ) is infinite. This provides a partial answer to a question asked by Johnson and Odell.