The consistency strength of certain stationary subsets of 𝒫_{𝜅}𝜆

Abstract

If κ ⩽ λ \kappa \leqslant \lambda are uncountable cardinals with κ \kappa regular, let S ( κ , λ ) S\left ( {\kappa ,\lambda } \right ) . We investigate the consistency strength of the statement " S ( κ , λ ) S\left ( {\kappa ,\lambda } \right ) is stationary in P κ λ {\mathcal {P}_\kappa }\lambda ," and prove that it is strictly weaker than " ∃ \exists a Ramsey cardinal," which combines with the lower bound ( 0 # ) \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 ( L [ U ] ) \left ( {L\left [ U \right ]} \right ) to show that " ∃ λ > \exists \lambda > " does not necessarily imply " S ( κ , κ + ) S\left ( {\kappa ,{\kappa ^ + }} \right ) is stationary."