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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 $\operatorname{cf}( \lambda ) < \lambda = \aleph_{\lambda}$.

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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 $\lambda^+$.

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  • $\begingroup$ Thank you for your answer and your attention! Yes, SSH ($\operatorname{pp} (\lambda) = \lambda^+$ for every singular $\lambda$) follows from GCH and from "$0^{\sharp}$ does not exist", and SSH implies that $\operatorname{cov} ( \lambda , \lambda, {(\operatorname{cf}(\lambda))}^{+} , 2) = \lambda^+$ for every singular $\lambda$. $\endgroup$ Aug 26, 2013 at 3:28
  • $\begingroup$ I edited my question to make it more clear. $\endgroup$ Aug 27, 2013 at 19:38
  • $\begingroup$ I'll edit my answer to address your revised question as soon as I get a little time. First week of semester is always busy. $\endgroup$ Aug 29, 2013 at 1:15
  • $\begingroup$ The link in this answer no longer works - here is at least a Wayback Machine snapshot: main page and publications. $\endgroup$ Dec 29, 2022 at 7:46

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