Combinatorial characterization of $\Pi^1_1$-indescribability in $P_{\kappa}\lambda$

Abstract.

It is proved that $\Pi^1_1$-indescribability in $P_{\kappa}\lambda$ can be characterized by combinatorial properties without taking care of cofinality of $\lambda$. We extend Carr's theorem proving that the hypothesis $\kappa$ is $2^{\lambda^{<\kappa}}$-Shelah is rather stronger than $\kappa$ is $\lambda$-supercompact.