Saturated filters at successors of singulars, weak reflection and yet   another weak club principle
  Suppose that lambda is the successor of a singular cardinal mu whose cofinality is an uncountable cardinal kappa. We give a sufficient condition that the club filter of lambda concentrating on the points of cofinality kappa is not lambda^+-saturated. The condition is phrased in terms of a notion that we call weak reflection. We discuss various properties of weak reflection 
Mirna D\v{z}amonja, Saharon Shelah