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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 exist infinite $M = (m_i)_{i=1}^∞, N ⊆ ℕ$ such that $ℱ[N](M) = {{m_i:i ∈ F}:F ∈ ℱ and F ⊂ N } ⊆ S_α$.