Asymptotic models via plegma families

Abstract

It is known that there exists a Banach space $X$ with a Schauder basis $(e_i)_{i=1}^{\infty}$ which does not admit $\ell_p$ as the model space obtained by a finite chain of sequences such that each element is a spreading model of a block subsequence of the previous element, starting from a block subsequence of $(e_i)_{i=1}^{\infty}$. We prove that $X$ has the stronger property of not admitting $\ell_p$ via a finite chain consisting of block asymptotic models. This is related to a question posed by L. Halbeisen and E. Odell for the special case of block generated asymptotic models. Also, we show that for every $k \in \mathbb{N}$ the Ramsey Coloring Theorem for $[\mathbb{N}]^{k}$ is equivalent to the following $k$-oscillation stability: In an arbitrary Banach space $X$, for every $\epsilon > 0$ and for every normalized sequence $(e_i)_{i \in \mathbb{N}}$ in $X$ there exists $M \in [\mathbb{N}]^{\infty}$ such that if $n_1,n_2,\cdots,n_k,m_1,m_2,\cdots,m_k \in M$, then $$ \big| ||\sum_{i =1}^{k}a_i e_{n_i} || - ||\sum_{i=1}^{k} a_i e_{m_i} || \big| < \epsilon $$ for all $(a_i)_{i=1}^{k} \in [-1,1]^k$.