Kruskal-Katona Theorem with Majority?
I am not sure if these results are known. If you define the $t$-thick upper (or lower, does not matter) shadow of $A_k$ as the sets of one level higher (resp. lower) that contain (resp. are contained in) at least $t$ sets from $A_k$, then what you ask for is the $(k+1)/2$-thick upper shadow. As far as I know it is open to determine the size of the smallest $A_k$ whose $2$-thick shadow is empty. Unfortunately I cannot point you to a reference but I think I heard it from Katona. (And in case I am wrong, then I remember incorrectly...)