A well-known application of the Ramsey Theorem is the proof of the Brunel–Sucheston Theorem. Based on this application, as an intermediate step, we consider the concept of \((k,\varepsilon )\)-oscillation stable sequence, which is generalized with the notion of \(((\mathcal {B}_i)_{i=1}^k,\varepsilon )\)-block oscillation stable sequence where \((\mathcal {B}_i)_{i=1}^k\) is a finite sequence of barriers on \(\mathbb {N}\). We prove that the Ramsey Theorem is equivalent (i.e., one result is deduced by using the other) 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 exists 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. We also prove a theorem like the Brunel–Sucheston Theorem which introduces the notion of \((\mathcal {B}_i)_{i\in \mathbb {N}}\)-block asymptotic model of a normalized basic sequence where \((\mathcal {B}_i)_{i\in \mathbb {N}}\) is a sequence of barriers. This notion includes the notion of spreading model of Brunel–Sucheston and we provide an example of a block asymptotic model that is not a spreading model. Moreover, we observe that some of our main results are equivalent to the Ramsey Theorem.

拉姆西定理的一个著名应用是布鲁内尔-休斯顿定理的证明。基于此应用，作为中间步骤，我们考虑\（（k，\ varepsilon）\）-振荡稳定序列的概念，该概念用\（（（\ mathcal {B} _i）_ { i = 1} ^ k，\ varepsilon）\） -块振荡稳定序列，其中\（（\ mathcal {B} _i）_ {i = 1} ^ k \）是\（\ mathbb { N} \）。我们证明了拉姆西定理与该语句是等价的（即，一个结果可通过使用另一个推论得出）：

• “对于 障碍的每个有限序列\（（\ mathcal {B} _i）_ {i = 1} ^ k \） ，每个 \（\ varepsilon> 0 \） 和每个规范化序列 \（（x_i）_ {i \在\ mathbb {N}} \） 中存在一个子序列 \（（x_i）_ {i \ in M} \） 是 \（（（\ mathcal {B} _i \ cap \ mathcal {P}（M）） _ {i = 1} ^ k，\ varepsilon）\） -稳定的振荡”，

其中\（\ mathcal {P}（M）\）是无限集M的幂集。我们还证明了一个类似Brunel–Sutston定理的定理，它引入了\（（\ mathcal {B} _i）_ {i \ in \ mathbb {N}} \）的概念-归一化基本序列的块渐近模型，其中\ （（\ mathcal {B} _i）_ {i \ in \ mathbb {N}} \}是障碍的序列。该概念包括Brunel–Sutston的扩展模型的概念，我们提供了一个非渐进模型的块渐近模型的示例。此外，我们观察到我们的一些主要结果等同于拉姆西定理。