  Employing a construction of Tsirelson-like spaces due to Argyros and
Deliyanni, we show that the class of all Banach spaces which are isomorphic to
a subspace of $c_{0}$ is a complete analytic set with respect to the Effros
Borel structure of separable Banach spaces. Moreover, the classes of all
separable spaces with the Schur property and of all separable spaces with the
Dunford-Pettis property are $\mathbf{\Pi}^{1}_{2}$-complete.
