This is an announcement for the paper "Finite forms of Gowers' theorem on the oscillation stability of $c_0$" by Diana Ojeda-Aristizabal. Abstract: We give a constructive proof of the finite version of Gowers' $FIN_k$ Theorem and analyse the corresponding upper bounds. The $FIN_k$ Theorem is closely related to the oscillation stability of $c_0$. The stabilization of Lipschitz functions on arbitrary finite dimensional Banach spaces was studied well before by V. Milman. We compare the finite $FIN_k$ Theorem with the finite stabilization principle in the case of spaces of the form $\ell_{\infty}^n$, $n\in\mathbb{N}$ and establish a much slower growing upper bound for the finite stabilization principle in this particular case. Archive classification: math.CO math.FA Mathematics Subject Classification: 05D10 Remarks: 18 pages Submitted from: dco34@cornell.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1312.4639 or http://arXiv.org/abs/1312.4639