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A weak version of AT from OCA. (English) Zbl 0824.04003

Set theory of the continuum, Pap. Math. Sci. Res. Inst. Workshop, Berkeley/CA (USA) 1989, Math. Sci. Res. Inst. Publ. 26, 281-291 (1992).
[For the entire collection see Zbl 0758.00014.]
The paper is devoted to the study of the following statement, abbreviated WAT: Let \(J\subseteq P(\omega)\), \(M\subseteq P(\omega)\), \(A\subseteq P(\omega)\) and \(F: P(\omega)\to P(\omega)\) be as follows: (i) \(J\) is a proper ideal over \(\omega\) containing all finite sets; (ii) \(\bigcup_{a\in M} P(a)\subseteq M\) and for every \(b\in J\), \(M\cap \{b- n: n\in \omega\}\neq \emptyset\); (iii) \(A\) is an uncountable pairwise almost disjoint family with the following property: there is a 1-1 \(e: \omega\to 2^{<\omega}\) such that given \(a\in A\) and \(n,m\in a\), either \(e(n)\subseteq e(m)\), or else \(e(m)\subseteq e(n)\); (iv) \((F(a)\cap F(b)) \Delta F(a\cap b)\in J\). Then for all but countably many \(a\in A\), one can find \(a_ i\), \(i\leq k\), where \(k\in \omega\) and \(\bigcup_{i\leq k} a_ i= a\), with the following property: there are Borel functions \(G_ n: P(a_ i)\to P(\omega)\), \(n< \omega\), such that \(\{F(b) \Delta G_ n(b): n\in \omega\}\cap \{c\cup d: c,d\in M\}\neq \emptyset\) for all \(b\in P(a_ i)\).
WAT is shown to be a consequence of the Open Coloring Axiom (OCA). Several statements are shown to follow from WAT (and thus from OCA), including this one: If \(f(n)= I\) and \(g(n)= {1\over n+ 1}\), then \(P(\omega)/J_ f\) and \(P(\omega)/J_ g\) are not isomorphic, where \[ J_ h= \left\{a\subseteq \omega: \limsup_{n\to \infty} {\sum_{m\in a\cap n} h(m)\over \sum_{m< n} h(m)}= 0\right\}. \] (\(P(\omega)/J_ f\) and \(P(\omega)/J_ g\) are known to be isomorphic under CH).
Reviewer: P.Matet (Caen)

MSC:

03E05 Other combinatorial set theory
03E35 Consistency and independence results

Citations:

Zbl 0758.00014
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