A proof of Shelah's strong covering theorem for \({\mathcal P}_{\kappa} \lambda\) (Q937818)

A theorem of \textit{S. Shelah} [``Advances in cardinal arithmetic'', in: N. W. Sauer et al. (eds.), Finite and infinite combinatorics. Dordrecht: Kluwer. NATO ASI Ser., Ser. C, Math. Phys. Sci. 411, 355--383 (1993; Zbl 0844.03028)] answered a long-standing open question in a surprising way: for every regular uncountable \(\kappa\) and \(\lambda>\kappa\), the smallest size of a stationary set in \(\mathcal P_\kappa\lambda\) is the same as the smallest size of an unbounded set in \(\mathcal P_\kappa\lambda\). By work of Baumgartner and Taylor, this was known in the situation when \(\lambda<\kappa^{+\omega}\), but the argument stopped there. Shelah's proof uses pcf theory and an ingenious system of elementary submodels. Unfortunately, close inspection of the details of this proof found some incorrect ones, as noticed by Shioya and apparently confirmed by Shelah. In this paper, Shioya gives the correct version of Shelah's proof, with the missing details pointed out and corrected. The historical notes in the introduction suggest that the corrections were suggested by Shelah in response to Shioya's inquiry and then further revised by Shioya. In addition to this proof, the paper gives a different, self-contained proof due to Shioya. The paper is well written and interesting to read. I was slightly annoyed by the use of \(\Pi A\) in place of \(\Pi A/U\) in the section reviewing pcf theory, making some of the statements quoted formally incorrect.