"cov vs pp" problem

Question

This is the problem $(\beta)$ of the section 14.7 in the "Analytical Guide" of the Shelah's book "Cardinal Arithmetic":

$(\beta)$ Is $\operatorname{cov} ( \lambda , \lambda, \aleph_{1} , 2) =^{+} \operatorname{pp} ( \lambda )$ when $\operatorname{cf}( \lambda ) = \aleph_0$?

Is this problem still open?

We know that $\operatorname{cov} ( \lambda , \lambda, {(\operatorname{cf}(\lambda))}^{+} , 2) = \operatorname{pp} (\lambda )$ for every singular $\lambda$ such that $\lambda < \aleph_\lambda$ (see for instance the Claim 3.7(1) in the Chapter IX of the Shelah's book).

Can the following statement be a theorem in ZFC?

$\operatorname{cov} ( \lambda , \lambda, {(\operatorname{cf}(\lambda))}^{+} , 2) = \operatorname{pp} (\lambda )$ for every infinite cardinal $\lambda$ with $\operatorname{cf}( \lambda ) < \lambda = \aleph_{\lambda}$.

Answer 1

Very much an open problem. I've been sporadically blogging about aspects of this question over the past few years, mainly in an effort to pin down exactly what's known and what's still unknown. There are some links to this work on my website.

With regard to your other question, the answer is yes under GCH, because both the covering number and the pp number must be equal to $\lambda^+$.