  Gowers' FIN$_k$ theorem, also called Gowers' pigeonhole principle or Gowers'
theorem, is a Ramsey-type theorem. It first occurred in the study of Banach
space theory and is a natural generalization of Hindman's theorem. In this
short note, we will show that Gowers' FIN$_k$ theorem does not follow from
ACA$_0$.
