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We show that the Schreier sets $S_{\alpha }(\alpha <{\omega }_1)$ have the following dichotomy property. For every hereditary collection ℱ of finite subsets of ℱ, either there exists infinite $M={\left(m_i\right)}_{i=1}^{\infty }\subseteq \mathbb{N}$ such that $S_{\alpha }\left(M\right)=m_i:i\in E:E\in S_{\alpha }\subseteq ℱ$, or there exist infinite $M={\left(m_i\right)}_{i=1}^{\infty },N\subseteq \mathbb{N}$ such that $ℱ\left[N\right]\left(M\right)=m_i:i\in F:F\in ℱandF\subset N\subseteq S_{\alpha }$.