A dichotomy on Schreier sets

Abstract

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 $\mathbb{N}$, either there exists infinite $M=(m_i{)}_1^{\infty }\subseteq \mathbb{N}$ such that $\cs_α(M)=\{\{m_i:i\in E\}:E\in\cs_α\}\subseteq\cf$, or there exist infinite $M=(m_i{)}_1^{\infty },N\subseteq \mathbb{N}$ such that $\cf[N](M)=\{\{m_i:i\in F\}:F\in\cf \mbox{ and } F\subset N\}\subseteq\cs_α$.

References in corpus (1)

• Complexity of weakly null sequences