Families of finite subsets of \(\mathbb{N}\) of low complexity and Tsirelson type spaces (Q2711598)

In this article the authors study Tsirelson type spaces defined by finite sequences \(({\mathcal M}_k)_{k=1}^{l}\) of compact families of finite subsets of \(\mathbb N\), where the families \({\mathcal M}_k\) are of low complexity. Using an appropriate index, denoted by \(i({\mathcal M})\), to measure the complexity of a family \({\mathcal M}\), they prove :NEWLINENEWLINENEWLINE(i). If \(i({\mathcal M})< \omega\) for all \(k= 1,\dots , l\) then the space \(T[({\mathcal M}_k)_{k=1}^{l}]\) contains isomorphically some \(\ell_p\), \(1<p< \infty\), or \(c_0\).NEWLINENEWLINENEWLINE(ii). If \(i({\mathcal M})= \omega\), then the space \(T[({\mathcal M}_k)_{k=1}^{l}]\) contains a subspace isomorphic to a subspace of the original Tsirelson's space.