The problem from the title is well-known and has been around since 1971: is every normal, locally compact, metacompact space paracompact? Arkhangel’skij proved that ‘perfectly normal’ suffices. In this paper the authors produce a model of ZFC in which there is a counterexample, thus laying the problem to rest. The problem is tucked in even tighter in another paper [the authors, ibid. 149, 275-285 (1996; Zbl 0855.54006)], where is shown that \(\text{MA}_{\sigma \text{-centered}} (\omega_1)\) implies a positive answer to the problem and that under the full \(\text{MA} (\omega_1)\) metacompactness may be relaxed to meta-Lindelöfness.
A brief description of the space follows: the underlying set is \(\omega_1\cup \bigcup_{n\in\omega} \omega_1^{\leq n}\). Each \(\omega_1^{\leq n}\) is given a compact Hausdorff topology by taking the family of sets of the form \(V(\sigma)= \{\tau: \sigma\subseteq \tau\}\) and their complements as a subbase. The sets \(\omega_1^{\leq n}\) will be clopen in the space. To get a local base at \(\alpha\in \omega_1\) one takes \(X_\alpha \subseteq \omega\) and, for each \(n\in X_\alpha\) an element \(\sigma_{\alpha,n}\) of \(\omega_1^{\leq n}\). A typical basic neighbourhood of \(\alpha\) then is \(\{\alpha\}\cup \bigcup_{n\in X_\alpha\smallsetminus k}V(\sigma_{\alpha,n})\). The demands are that (0) the \(X_\alpha\) form an almost disjoint family (to make the space Hausdorff); (1) for each \(A\subseteq \omega_1\) there must be \(u\subseteq \omega\) such that \(X_\alpha \subseteq^* u\) if \(\alpha\in A\) and \(X_\alpha\cap u=^*\emptyset\) if \(\alpha\not\in A\) (a step towards normality); (2) if \(\sigma\in \omega_1^{\leq n}\) then \(\{\alpha: \sigma_{\alpha,n}= \sigma\}\) is finite (towards metacompactness) and (3) if \(A\subseteq \omega_1\) then there are a countable subset \(B\) of \(A\), and a finite-to-one function \(f: B\to\omega\) such that for all \(\alpha\), for almost all \(n\in X_\alpha\) there is a \(\beta\in B\) such that \(n\in X_\beta\smallsetminus f(\beta)\) and \(\sigma_{\alpha,n}\supseteq \sigma_{\beta,n}\) (another step towards normality).
The bulk of the paper is devoted to showing that, consistently, such a choice is possible. The \(X_\alpha\) and \(\sigma_{\alpha,n}\) are obtained in a single-step forcing; the rest of the forcing is a proper countable-support iteration of length \(\omega_2\) in which condition 1 is made true, while retaining 3.