A theorem on \(P_{\kappa}(\lambda)\)
The set \(S(\kappa,\kappa^+)\) is \(\{x\in P_{\kappa}(\kappa^+)|| x| =| x\cap \kappa |^+\}\). It was investigated in \textit{S. Baldwin} [Proc. Am. Math. Soc. 92, 90-92 (1984; Zbl 0552.03035)]. Here it is shown that if f is an ordinal-valued function on \(\kappa\) with Galvin-Hajnal norm \(\| f\| =\beta\), where \(\beta <\lambda\), then the order-type OT(\(\beta\cap x)\leq f(OT(\kappa \cap x))\) for almost all \(x\in {\mathcal P}_{\kappa}(\lambda)\); as a corollary, if (i) \(S(\kappa,\kappa^+)\) is stationary then (ii) the function \(f(\xi)=\xi^+\) on \(\kappa\) has norm \(\kappa^+\). The paper ends with some remarks and open questions on the consistency strengths of (i) and (ii).
scientific article; zbMATH DE number 4029547