On the free subset property at singular cardinals (Q1114681)

Let S be a first-order structure; for \(X\subseteq S\) let S[X] be the substructure of S generated from X. Call a subset X of S free if for all \(x\in X\), \(x\not\in S[X\setminus \{x\}]\). Let \(Fr_{\mu}(\kappa,\lambda)\) denote the statement that every structure of size \(\geq \kappa\) and with at most \(\mu\) constants, functions and relations has a free subset of size at least \(\lambda\). It is known that if \(\kappa\) is the minimal cardinal such that \(Fr_{\omega}(\kappa,\omega_ 1)\) holds then \(\kappa \geq \omega_{\omega_ 1}\), and \(\kappa\) is regular or of cofinality \(\omega_ 1\). The author shows that if this minimal \(\kappa\) is singular then there is an inner model with \(\omega_ 1\) measurable cardinals. The paper contains a full proof that an inner model with at least one measurable cardinal exists under the above assumption, preceded by a good sketch. Basically the author shows that the statements ``no inner model with a measurable'' and ``the structure \((H_{\kappa^+})^ K\) has an uncountable free subset'' are incompatible; here K is the core model. The proof of the main result is oulined; it relies on the theory of short core models, as presented by \textit{P. Koepke} [Ann. Pure Appl. Logic 37, 179-204 (1988; Zbl 0638.03049)].