The Ramsey theorem in Banach spaces

The Ramsey theorem in Banach spaces was first proved by Field Medalist T.W. Gowers in 2002. Due to the complexity of its proof, people have discovered several other proofs since then, so the rest of the world can understand its proof. I will present a simple proof of this theorem, (or at least I try!), and I may even try to do it in Greek! I shared the fun for discovering this proof with N. Kalton and S. Dilworth.