Covering for category and combinatorics on \(P_\kappa(\lambda)\) (Q819562)

For \(\lambda>\kappa>\omega\), set \(A\in I_{\kappa,\lambda}\) if \(A\subseteq [\lambda]^{<\kappa}\) and there is an \(a\in [\lambda]^{<\kappa}\) such that for no \(x\supseteq a\) does \(x\in A\) hold. Various properties of the ideal \(I_{\kappa,\lambda}\) are investigated. For example, it is consistent (relative to the existence of a supercompact cardinal) that \(I^+_{\kappa,\kappa^+}\to (I^+_{\kappa,\kappa^+})^n\) (\(n<\omega\)) and \(\kappa\) is the least measurable. Conditions are given for \(I^+_{\kappa,\lambda}\to(I^+_{\kappa,\lambda},\theta)^2\) in terms of ineffability, existence of Kurepa families, and cardinal invariants of \({}^\lambda 2\). It is also described when any two cofinal subsets of \([\lambda]^{<\kappa}\) contain isomorphic subsets.