The family \({\mathcal A}\) of some countable subsets of the infinite cardinal \(\kappa\) is said to be saturated (or completely separable) if for all subsets x of \(\kappa\) either there is \(a\in {\mathcal A}\) with \(a\subseteq x\) or else there are finitely many sets \(a_ 1,...,a_ n\in {\mathcal A}\) with \(x-(a_ 1\cup...\cup a_ n)\) finite. Let S(\(\kappa\)) be the statement: there exists an infinite, almost disjoint, saturated family of countable subsets of the cardinal \(\kappa\). The existence of saturated families is a long-standing problem in set theory. In the early seventies Baumgartner showed that \(\forall nS(\aleph_ n)\) is a consequence of the Continuum Hypothesis. Later, \textit{A. Hajnal, I. Juhász} and \textit{L. Soukup} [Comment. Math. Univ. Carol. 28, 629-633 (1987; Zbl 0648.03034)] produced a forcing model to show the relative consistency of \(\forall \kappa S(\kappa)\). In this paper, the authors show that \(\forall \kappa S(\kappa)\) is a consequence of \(V=K\), where K is the core model. (In fact, they give more technical conditions under which their induction will go through.)