Finite high-order games and an inductive approach towards Gowersʹs dichotomy Original Research Article

We present the notion of finite high-order Gowers games, and prove a statement parallel to Gowersʹs Combinatorial Lemma for these games. We derive ‘quantitative’ versions of the original Gowers Combinatorial Lemma and of Gowersʹs Dichotomy, which place them in the context of the recently introduced infinite dimensional asymptotic theory for Banach spaces.