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Delta-System Lemma counterexample

  1. Nov 20, 2009 #1
    The [tex]\Delta[/tex]-system lemma states the following: given an infinite cardinal [tex]\kappa[/tex], let [tex]\theta > \kappa[/tex] be a regular cardinal such that [tex]\forall \alpha < \theta \ (|\alpha^{< \kappa}| < \theta)[/tex]; given [tex]A[/tex] such that [tex]|A| \geq \theta[/tex] and [tex]\forall x \in A \ (|x| < \kappa)[/tex], then there is a [tex]B \subset A[/tex] which forms a [tex]\Delta[/tex]-system, such that [tex]|B| = \theta[/tex].

    I wish to find a counterexample if [tex]\kappa[/tex] is not regular. For example, I wish to prove that there is a family [tex]A[/tex] with [tex]|A| = \omega_\omega[/tex] and [tex]\forall x \in A \ (|x| < \omega)[/tex] such that no [tex]B \subset A[/tex] (with [tex]|B| = \omega_\omega[/tex]) is a [tex]\Delta[/tex]-system.

    Every help is appreciated. Thanks.
  2. jcsd
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