A dichotomy on Schreier sets

Abstract

We show that the Schreier sets $S_α(α < ω_1)$ have the following dichotomy property. For every hereditary collection ℱ of finite subsets of ℱ, either there exists infinite $M = (m_i)_{i=1}^∞ ⊆ ℕ$ such that $S_α(M)={{m_i:i ∈ E}: E ∈ S_α} ⊆ ℱ$, or there exi