More canonical forms and dense free subsets. (Q1428037)

Answering an old question of Komjáth, and extending some deep results of Devlin, Shelah, and Koepke, the author shows that the consistency of ``ZFC+GCH+there are infinitely many compact cardinals'' implies the consistency of: ZFC+GCH+every structure on \(\aleph_\omega\) with countable signature has a free subset that contains one point in \([\aleph_n,\aleph_{n+1})\) for each \(n\geq 0\). The core of the proof is (an extension of) a theorem asserting, under the same consistency assumption, the consistency of: ZFC+GCH+for each sequence \(\langle f_n : n\in\omega\rangle\) with each \(f_n\) being \(n\)-ary and regressive on \(\aleph_\omega\), there exist sets \(S_n\subseteq[\aleph_n,\aleph_{n+1})\), \(n>0\), such that for each \(n>0\), \(| S_n| \geq\aleph_{n-1}\) and \(f_n\) is constant on \(\prod_{i=1}^nS_i\). The methods include working with indiscernibles and carefully forcing with Levy collapses.