The consistency strength of certain stationary subsets of $\mathcal {P}_\kappa \lambda$

Abstract:

If $\kappa \leqslant \lambda$ are uncountable cardinals with $\kappa$ regular, let $S\left ( {\kappa ,\lambda } \right )$. We investigate the consistency strength of the statement "$S\left ( {\kappa ,\lambda } \right )$ is stationary in ${\mathcal {P}_\kappa }\lambda$," and prove that it is strictly weaker than "$\exists$ a Ramsey cardinal," which combines with the lower bound $\left ( {{0^\# }} \right )$ proven earlier by J. Baumgartner to give a narrow range of the consistency strength of this statement. In addition, we give an example $\left ( {L\left [ U \right ]} \right )$ to show that "$\exists \lambda >$" does not necessarily imply "$S\left ( {\kappa ,{\kappa ^ + }} \right )$ is stationary."