# Delta-System Lemma counterexample

1. Nov 20, 2009

### eurialo

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

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

Every help is appreciated. Thanks.