The connection between cardinalities of formular subsets and the stability of formulas (Q1090328)

Let T be a theory. The author calls a formula \(\phi\) (x) stable in T if \(R_ T(\phi,\Delta,2)<\infty\) (Shelah rank) for every finite \(\Delta\). He proves that for every stable formula \(\phi_ 1(x)\) and any formula \(\phi_ 0(x)\), if in every \({\mathfrak A}\vDash T\) there holds \(card(\phi_ 0({\mathfrak A}))\leq card(\phi_ 1({\mathfrak A}))\), then \(\phi_ 0\) is T- stable, too. The main part of the proof seems to be contained in a lemma of the author's paper in Algebra Logika 14, 245-257 (1975; Zbl 0331.02034). The other part uses a pigeon-hole principle to show how instability goes down. There are a lot of misprints (most of them are already in the original paper) complicating the understanding of the paper considerably.