A dichotomy on Schreier sets

ArticleOriginal scientific text

Title

Authors ¹

Department of Mathematics, Oklahoma State University, Stillwater, Oklahoma 74078-0613, U.S.A.

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}^{\infty} \subseteq ℕ

such that

S_{α} (M) = {{m_{i} : i \in E} : E \in S_{α}} \subseteq ℱ

, or there exist infinite

M = {(m_{i})}_{i = 1}^{\infty}, N \subseteq ℕ

such that

ℱ [N] (M) = {{m_{i} : i \in F} : F \in ℱ and F \subset N} \subseteq S_{α}

.

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