The paper is about polarized partition relations of the form \(\binom{\lambda}{\kappa}\to\binom{\lambda}{\kappa}_\theta\) where this means the following. For every map \(c:\lambda\times\kappa\to\theta\) there exist sets \(A\in[\lambda]^{\lambda}\) and \(B\in[\kappa]^{\kappa}\) and an ordinal \(i\in\theta\) such that the direct image \(c[A\times B]=\{i\}\). This relation is called strong, and the following \(\binom{\lambda}{\kappa}\to\binom{\tau}{\kappa}_\theta\), is almost strong, where this relation assures that for every map \(c:\lambda\times\kappa\to\theta\) and any ordinal \(\tau<\lambda\), there are sets \(A\subseteq\lambda\) and \(B\subseteq{\kappa}\) and an ordinal \(i<\theta\) such that \(c[A\times B]=\{i\}\), where \(\operatorname{otp}(A)=\tau\), \(\operatorname{otp}(B)=\kappa\). The aim of the paper is to obtain the almost strong relation for limit singular cardinals, which generalizes previous results of P. Erdős et al. [Acta Math. Acad. Sci. Hung. 16, 93–196 (1965; Zbl 0158.26603)] and S. Shelah [Fundam. Math. 155, No. 2, 153–160 (1998; Zbl 0897.03050)]. While those improvements would be enough to justify the publication of this work, the authors develop a novel and amazing method using pcf theory to construct the main tool to prove their results.
Let us first review previous achievements. Erdős, Hajnal, Rado [loc. cit.] have proved \(\binom{\mu^+}{\mu}\to\binom{\mu}{\mu}_2\) when \(\operatorname{cf}(\mu)=\omega\). In the paper under review, the authors show the same relation but they increase the order type of the first coordinate, that is, \(\operatorname{otp}(A)=\tau\) for every \(\tau<\mu^+\). On the other hand, Shelah shows that \(\binom{\mu^+}{\mu}\to\binom{\mu+1}{\mu}_{<\operatorname{cf}(\mu)}\) whenever \(\mu>\operatorname{cf}(\mu)>\omega\), \(\mu\) is a strong limit and \(2^\mu>\mu^+\). The authors reach the same result replacing \(\mu+1\) by any ordinal \(\tau<\mu^+\) for singular cardinals of cofinality \(>\omega\) and \(2^\mu>\mu^+\).
In order to prove the mentioned results, the authors introduce the notion of an pcf array of elementary submodels, which is a matrix \((M_{\alpha i}:\alpha<\lambda, i<\kappa)\). Every model is an elementary substructure of \(H_\eta\) for \(\eta\) big enough, the rows are increasing elementary chains, the columns are also increasing elementary chains but only when \(i\) does not belongs to some \(u\in J\), where \(J\) is an ideal containing the ideal of bounded subsets of \(\kappa\). There exists a sequence \(l=(\lambda_i:i<\kappa)\), with \(\mu=\bigcup_{i<\kappa}\lambda_i\) and \(\kappa=\operatorname{cf}(\mu)\). Moreover, \(|M_{\alpha i}|<\lambda_i\), where \(\mu<\lambda<\operatorname{tcf}\left(\prod_{i<\kappa}\lambda_i,J\right)\). We are not going into the details of the pcf constructions and related notions (detailed explanations can be read in the article).
Given such a matrix \(\mathcal{M}\), for every \(\alpha<\lambda\), \(i<\kappa\) let \(f_\alpha(i)=\sup(M_{\alpha i}\cap \lambda_i)\). Each \(f_\alpha\) is called the characteristic function of \(\mathcal{M}\) at stage \(\alpha\) and \((f_\alpha:\alpha<\lambda)\) is the characteristic sequence. As a consequence of pcf theory, there is a function \(h\in\prod_{i<\kappa}\lambda_i\) which bounds the sequence. Under certain hypotheses, the authors manage to construct such an array \(\mathcal{M}\), which is the key to prove the above mentioned results.
In some sense, the matrix works as some kind of morass, but outside of \(V=L\).