Saturated filters at successors of singulars, weak reflection and yet another weak club principle

Suppose that λ is the successor of a singular cardinal μ whose cofinality is an uncountable cardinal κ. We give a sufficient condition that the club filter of λ concentrating on the points of cofinality κ is not λ⁺-saturated.¹ The condition is phrased in terms of a notion that we call weak reflection. We discuss various properties of weak reflection. We introduce a weak version of the Clubsuit sign-principle, which we call Clubsuit sign*_-, and show that if it holds on a stationary subset S of λ, then no normal filter on S is λ⁺-saturated. Under the above assumptions, Clubsuit sign*_-(S) is true for any stationary subset S of λ which does not contain points of cofinality κ. For stationary sets S which concentrate on points of cofinality κ, we show that Clubsuit sign*_-(S) holds modulo an ideal obtained through the weak reflection.