  W. T. Gowers proved that every Lipschitz function from the unit sphere of the
Banach space $c_0$ to $\mathbb{R}$ is oscilation stable. His proof uses a
result about finite partitions of the set $FIN_k$ of finitely supported
functions $p$ from $\mathbb{N}$ to $\{0,1,...,k\}$ with $k$ in $Im(p)$. Every
known proof of this fact uses methods of topological dynamics on the space
$\beta\mathbb{N}$ of ultrafilters on $\mathbb{N}$. We give a purely
combinatorial proof of this result avoiding the use of ultrafilters.
