Forcing many positive polarized partition relations between a cardinal and its powerset (Q2758064)

For an ordinal \(\sigma\) and a cardinal \(\tau\), let \(\text{FP}(\sigma,\tau,n)\) denote the set of finite partial functions from \(\sigma\) to \(\tau\) with domain of size \(n\). For cardinal numbers \(\tau\), \(\kappa\) and \(\chi\), the polarized partition symbol \(((\tau)_\sigma) \rightarrow ((\kappa)_\sigma)_\chi^{((1)_n)}\) means that whenever \(F: \text{FP}(\sigma,\tau,n) \to \chi\), there exist \(Y_i \subseteq \tau\), \(i < \sigma \), of size \(\kappa\) such that for all \(a,b \in \text{FP} (\sigma,\tau,n)\), if \(\text{dom}(a) =\text{dom}(b)\) and \(a(i), b(i)\in Y_i\) for all \(i \in \text{dom}(a)\), then \(F(a) = F(b)\). NEWLINENEWLINENEWLINEThe authors prove that, assuming GCH, if \(\lambda < \mu\) are regular cardinals, then there is a \(\lambda\)-closed cofinality-preserving poset \(P\) forcing that \(2^\lambda = \mu\) and that for \(\sigma < \lambda \leq \kappa < \chi < \tau \leq \mu\) such that \(\chi\) and \(\tau\) are sufficiently far apart, \(((\tau)_\sigma) \rightarrow ((\kappa)_\sigma)_\chi^{((1)_n)}\) holds, thus generalizing an earlier result of the first author [Isr. J. Math. 62, 355-380 (1988; Zbl 0657.03028)]. NEWLINENEWLINENEWLINEThe forcing adds \(\mu\) Cohen subsets of \(\lambda\) with a kind of mixed support and thus is an elaboration of the method in the latter work. The main point of the forcing is that \(p<q\) decomposes in a pure extension which has closure properties and an apure extension which has local chain condition properties so that preservation of cofinalities is guaranteed. The proof of the partition relation uses that \(\tau \rightarrow (\kappa)_\chi^{m(n)}\) holds in the ground model for a sufficiently large \(m(n)\), and then appeals to a result of the first author [Set theory and its applications, Lect. Notes Math. 1401, Springer, Berlin, 167-193 (1989; Zbl 0683.04002)].