Order types of free subsets (Q1377636)

Let \(\kappa\) be a cardinal, \(\alpha\) and \(\beta\) ordinals. \(\text{Fr}^{\text{ord}}_\kappa (\alpha,\beta)\) denotes the following property: For any \(\tau\cup \{<\}\)-structure \({\mathcal A}= (A,<^A, \dots)\) with \(|\tau|\leq \kappa\) and \((A,<^A) \cong (\alpha,\in)\) there is a free subset \(S\) of \(A\) of order type \(\beta\). Fixing \(\kappa\), let \(\alpha (\beta)\) denote the least \(\alpha\) with \(\text{Fr}^{\text{ord}}_\kappa (\alpha,\beta)\). The author determines lower bounds for \(\alpha(\beta)\) and shows that given enough measurable cardinals, there are forcing extensions where the given bounds are sharp.