  A well-known application of the Ramsey Theorem in the Banach Space Theory is
the proof of the fact that every normalized basic sequence has a subsequence
which generates a spreading model (the Brunel-Sucheston Theorem). Based on this
application, as an intermediate step, we can talk about the notion of
$(k,\varepsilon)-$oscillation stable sequence, which will be described and
analyzed more generally in this article. Indeed, we introduce the notion
$((\mathcal{B}_i)_{i=1}^k,\varepsilon)-$block oscillation stable sequence where
$(\mathcal{B}_i)_{i=1}^k$ is a finite sequence of barriers and using what we
will call blocks of barriers. In particular, we prove that the Ramsey Theorem
is equivalent to the statement ``for every finite sequence
$(\mathcal{B}_i)_{i=1}^k$ of barriers, every $\varepsilon>0$ and every
normalized sequence $(x_i)_{i\in\mathbb{N}}$ there is a subsequence
$(x_i)_{i\in M}$ that is
$((\mathcal{B}_i\cap\mathcal{P}(M))_{i=1}^k,\varepsilon)-$block oscillation
stable'', where $\mathcal{P}(M)$ is the power set of the infinite set M.
Besides, we introduce the $(\mathcal{B}_i)_{i\in\mathbb{N}}-$block asymptotic
model of a normalized basic sequence where $(\mathcal{B}_i)_{i\in\mathbb{N}}$
is a sequence of barriers. These models are a generalization of the spreading
models and are related to the $((\mathcal{B}_i)_{i=1}^k,\varepsilon)-$block
oscillation stable sequences. We show that the Brunel-Sucheston is satisfied
for the $(\mathcal{B}_i)_{i\in\mathbb{N}}-$block asymptotic models, and we also
prove that this result is equivalent to the Ramsey Theorem. The difference
between our theorem and the Brunel-Sucheston Theorem is based on the number of
different models that are obtained from the same normalized basic sequence
through them. This and other observations about
$(\mathcal{B}_i)_{i\in\mathbb{N}}-$block asymptotic models are noted in an
example at the end of the article.

Functional Analysis;Combinatorics