# A dichotomy on Schreier sets

Studia Mathematica (1999)

• Volume: 132, Issue: 3, page 245-256
• ISSN: 0039-3223

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## Abstract

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We show that the Schreier sets ${S}_{\alpha }\left(\alpha <{\omega }_{1}\right)$ have the following dichotomy property. For every hereditary collection ℱ of finite subsets of ℱ, either there exists infinite $M={\left({m}_{i}\right)}_{i=1}^{\infty }\subseteq ℕ$ such that ${S}_{\alpha }\left(M\right)={m}_{i}:i\in E:E\in {S}_{\alpha }\subseteq ℱ$, or there exist infinite $M={\left({m}_{i}\right)}_{i=1}^{\infty },N\subseteq ℕ$ such that $ℱ\left[N\right]\left(M\right)={m}_{i}:i\in F:F\in ℱandF\subset N\subseteq {S}_{\alpha }$.

## How to cite

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Judd, Robert. "A dichotomy on Schreier sets." Studia Mathematica 132.3 (1999): 245-256. <http://eudml.org/doc/216598>.

@article{Judd1999,
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 $ℱ[N](M) = \{\{m_i:i ∈ F\}:F ∈ ℱ and F ⊂ N \} ⊆ S_α$.},
author = {Judd, Robert},
journal = {Studia Mathematica},
keywords = {Tsirelson type Banach spaces; Schreier sets; transfinite induction},
language = {eng},
number = {3},
pages = {245-256},
title = {A dichotomy on Schreier sets},
url = {http://eudml.org/doc/216598},
volume = {132},
year = {1999},
}

TY - JOUR
AU - Judd, Robert
TI - A dichotomy on Schreier sets
JO - Studia Mathematica
PY - 1999
VL - 132
IS - 3
SP - 245
EP - 256
AB - 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 $ℱ[N](M) = {{m_i:i ∈ F}:F ∈ ℱ and F ⊂ N } ⊆ S_α$.
LA - eng
KW - Tsirelson type Banach spaces; Schreier sets; transfinite induction
UR - http://eudml.org/doc/216598
ER -

## References

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1. [AA] D. Alspach and S. Argyros, Complexity of weakly null sequences, Dissertationes Math. 321 (1992).
2. [AO] D. Alspach and E. Odell, Averaging weakly null sequences, Lecture Notes in Math. 1332, Springer, 1988, 126-144. Zbl0668.46011
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5. [FJ] T. Figiel and W. B. Johnson, A uniformly convex Banach space which contains no ${\ell }_{p}$, Compositio Math. 29 (1974), 179-190. Zbl0301.46013
6. [KN] P. Kiriakouli and S. Negrepontis, Baire-1 functions and spreading models of ${\ell }_{1}$, preprint.
7. [OTW] E. Odell, N. Tomczak-Jaegermann and R. Wagner, Proximity to ${\ell }_{1}$ and distortion in asymptotic ${\ell }_{1}$ spaces, preprint.
8. [Sch] J. Schreier, Gegenbeispiel zur Theorie der schwachen Konvergenz, Studia Math. 2 (1930), 58-62. Zbl56.0932.02
9. [T] B. S. Tsirelson, Not every Banach space contains ${\ell }_{p}$ or ${c}_{0}$, Functional Anal. Appl. 8 (1974), 138-141.

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