Let $S=\lbrace 1,2,3,\ldots, 2^n\rbrace$ for some $n\ge 2$. What is the maximum cardinality of a set $S'$ of $2^{n-1}$-element subsets of $S$ such that every pair of elements of $S'$ has exactly $2^{n-2}$ elements in common?

The answer might be $2^n - 1$. This problem came up a while ago while I was working on something larger. I skirted around it, but it always bugged me.