  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 $(a_i)_{i=1}^{k} \in [-1,1]^k$.
