A dichotomy on Schreier sets

Abstract

We show that the Schreier sets $\mathcal{S}_α\ (α<ω_1)$ 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 $\cs_α(M)=\{\{m_i:i\in E\}:E\in\cs_α\}\subseteq\cf$, or there exist infinite $M=(m_i)_1^{\infty},N\subseteq\N$ such that $\cf[N](M)=\{\{m_i:i\in F\}:F\in\cf \mbox{ and } F\subset N\}\subseteq\cs_α$.