Ramsey property and block oscillation stability on normalized sequences in Banach spaces

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