Generalizing a result of Todorčević, we prove the existence of directed sets D, E such that D ≥ Pκλ and E ≥ Pκλ but D × E ≥ Pκλ in the Tukey ordering. As an application, we show that the tree property for directed sets introduced by Hinnion is not preserved under products.
A partition property of a mixed type for Pκ(λ)
✍ Scribed by Pierre Matet
- Publisher
- John Wiley and Sons
- Year
- 2003
- Tongue
- English
- Weight
- 249 KB
- Volume
- 49
- Category
- Article
- ISSN
- 0044-3050
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✦ Synopsis
Abstract
Given a regular infinite cardinal κ and a cardinal λ > κ, we study fine ideals H on P~κ~(λ) that satisfy the square brackets partition relation $ H^{+} {\mathop \rightarrow \limits^{\kappa}} [H^{+}]^{2}_{\mu} $, where μ is a cardinal ≥2. (© 2003 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)
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