  We show that the Schreier sets $\mathcal{S}_{\alpha}\ (\alpha<\omega_1)$
satisfy the following dichotomy property. For every hereditary collection $\cf$
of finite subsets of $\N$, either there exists infinite
$M=(m_i)_1^{\infty}\subseteq\N$ such that $\cs_{\alpha}(M)=\{\{m_i:i\in
E\}:E\in\cs_{\alpha}\}\subseteq\cf$, or there exist infinite
$M=(m_i)_1^{\infty},N\subseteq\N$ such that $\cf[N](M)=\{\{m_i:i\in F\}:F\in\cf
\mbox{ and } F\subset N\}\subseteq\cs_{\alpha}$.
