Katětov's problem (Q2781383)

This is an interesting and important paper. It shows that Katětov's problem has a consistent affirmative answer. Katětov proved that a compact space is metrizable if its cube is hereditarily normal (only if is trivial) and asked whether hereditary normality of the square would suffice. Consistent counterexamples have been known for some time, see e.g., [\textit{G. Gruenhage} and \textit{P. J. Nyikos}, ibid. 340, No. 2, 563-586 (1993; Zbl 0817.54004)]. The authors present a model where all compact spaces with a hereditarily normal square are metrizable. The construction is interesting in its own right. It is possible to construct a model with a Souslin tree \(S\) in which Martin's Axiom for \(\aleph_1\) many dense sets holds for posets whose product with \(S\) is ccc [\textit{P. Larson}, J. Symb. Log. 64, No. 1, 81-98 (1999; Zbl 0926.03061)]. The final model is obtained by forcing with the Souslin tree over this model. One obtains a model in which a variation of the principle~\(\mathcal K_2\) from [\textit{S. Todorčević}, Partition problems in topology, Contemp. Math. 84 (1989; Zbl 0659.54001)] holds: if \([\omega_1]^2=K_0\cup K_1\) and whenever \(A,B\in[\omega_1]^{\aleph_1}\) there are uncountable \(A'\subseteq A\) and \(B'\subseteq B\) with \(\bigl\{\{\alpha,\beta\}:\alpha<\beta\), \(\alpha\in A'\), \(\beta\in B'\bigr\}\subseteq K_0\), then there is an uncountable \(K_0\)-homogeneous set. Using this principle one deduces that the existence of a nonmetrizable compact space with a hereditarily normal square entails the existence of an \(\aleph_1\)-sized \(Q\)-set of reals; on the other hand, after forcing with a Souslin tree there are no such \(Q\)-sets.