Let $\Sigma$ be a set of $|\Sigma| = n$ subsets of the universe $[n]$, each of size $k$, with the property that any two of these subsets intersect on at most one element. It is easy to see that the largest possible $k =: k^*$ (for suitable $n$) is achieved when $\Sigma$ is a finite projective plane.
My question is if any stability results are known about this, informally meaning that any $\Sigma$ that comes "close" to achieving $k^*$ must be "close" to a finite projective plane. There are a few reasonable formalizations of this concept. One possible example: perhaps whenever $k \ge \varepsilon k^*$ for some $\Sigma$, there exists a finite projective plane $P$ and an injective map $\phi : \Sigma \to P$ so that the average size of $|S \cap \phi(S)|$ over all $S \in \Sigma$ is at least $f(\varepsilon) \cdot k$. But anything interpreted as a stability result would be on topic here.
