A proof of W. T. Gowers' $c_0$ theorem

W. T. Gowers' $c_0$ theorem asserts that for every Lipschitz function $F: S_{c_0} \to \mathbb{R}$ and $\varepsilon > 0$, there exists an infinite-dimensional subspace $Y$ of $c_0$ such that the oscillation of $F$ on $S_Y$ is at most $\varepsilon$. The proof of this theorem has been reduced by W. T. Gowers to the proof of a new Ramsey type theorem. Our aim is to present a proof of the last result.