On the inexistence of Continuous Tree-Like Scales, and the Approachable Free Subset Property

In his PhD thesis, Luis Pereira has isolated two properties of sequences of regular cardinals (kappa_n)_n from Shelah's PCF theory, which are related to the possible consistency of  2^{aleph_omega} being greater or equal to aleph_{omega_1}, when aleph_omega is a strong limit cardinal.

The first property is the inexistence of a continuous tree-like scale on the product of regular cardinals kappa_n, n < omega. A scale <f_alpha : alpha < lambda> is said to be tree-like if for every alpha < beta and n < omega, if t_alpha(n) and t_\beta(n) are distinct, then so are t_alpha(m), t_beta(m) for all m >n.

The second and stronger assertion is the Approachable Free Subset Property (AFSP) which asserts that for almost every (i.e., modulo a club) internally approachable structure N of a sufficiently large H_\theta, of size |N| < kappa_m for some m, the set of supremums { delta^N_n = sup(N \cap kappa_n) : n < omega } has an infinite subset X which is free with respect to the functions in N. Namely, for every function g in N of some finite arity k, and every delta in X, delta does not belongs to the g-image of [ X - {delta} ]^k.

Gitik has shown that the existence of a sequence of regulars (kappa_n)_n for which there does not exist a continuous tree-like scale, is consistent relative to the existence of a measurable cardinal kappa of Mitchell order o(kappa) = kappa^{+2}.
In a recent join study with Dominik Adolf, we have shown that the existence of a sequence (kappa_n)_n for which the AFSP holds is consistent, and that both AFSP and the inexistence of continuous tree-like scales on some sequence (kappa_n)_n are equi-consistent with the existence of a singular cardinal kappa for which {o(mu) :  mu < kappa} is unbounded in kappa.

The goal of the talk is to present and discuss the two properties and their connection to PCF theory, and describe some of the central ideas in the recent results.