A lower bound on Gowers' FIN_k theorem

Abstract

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${}_0$.