  We develop a general framework for infinite-dimensional Ramsey theory with
and without pigeonhole principle, inspired by Gowers' Ramsey-type theorem for
block sequences in Banach spaces and by its exact version proved by Rosendal.
In this framework, we prove the adversarial Ramsey principle for Borel sets, a
result conjectured by Rosendal that generalizes at the same time his version of
Gowers' theorem and Borel determinacy of games on integers.
