Suppose $n$ is a positive integer. Let ${\cal C}$ be a set of subsets of $X:=\{1,\ldots,n\}$ with the following properties:

- all members of ${\cal C}$ contain at least $2$ elements, and $X\notin {\cal C}$; and
- $A\neq B\in {\cal C}$ implies $|A\cap B| = 1$.

A version of Fisher's inequality states that $|{\cal C}| \leq n$. There are short proofs relying on Linear Algebra. Is there a purely combinatorial proof of this statement?