Finite high-order games and an inductive approach towards Gowers's dichotomy

We present the notion of finite high-order Gowers games, and prove a statem ent parallel to Gowers's Combinatorial Lemma for these games. We derive 'qu antitative' versions of the original Gowers Combinatorial Lemma and of Gowe rs's Dichotomy, which place them in the context of the recently introduced infinite dimensional asymptotic theory for Banach spaces. (C) 2001 Elsevier Science B.V. All rights reserved.