  We study delta-points in Banach spaces $h_{\mathcal{A},p}$ generated by
adequate families $\mathcal A$ where $1 \le p < \infty$. In the case the
familiy $\mathcal A$ is regular and $p=1,$ these spaces are known as
combinatorial Banach spaces. When $p > 1$ we prove that neither
$h_{\mathcal{A},p}$ nor its dual contain delta-points. Under the extra
assumption that $\mathcal A$ is regular, we prove that the same is true when
$p=1.$ In particular the Schreier spaces and their duals fail to have
delta-points. If $\mathcal A$ consists of finite sets only we are able to rule
out the existence of delta-points in $h_{\mathcal{A},1}$ and Daugavet-points in
its dual.
  We also show that if $h_{\mathcal{A},1}$ is polyhedral, then it is either
(I)-polyhedral or (V)-polyhedral (in the sense of Fonf and Vesel\'y).
