Let $n$ be a fixed positive integer. Let $S$ be a subset of points in the plane satisfying these conditions:
- There does not exist $n$ lines in the plane such that every element of $S$ lies on at least one of them.
- For all $x\in S$, there exist $n$ lines in the plane such that every element of $S-x$ lies on at least one of these lines.
What is the maximum value of $|S|$?
I believe that the maximum is $\frac{(n+1)(n+2)}{2}$, achieved when $S$ is a triangular array with $n+1$ vertices on each side. However, I'm not sure how to prove this.
EDIT: the same question was asked here, but no solution has been given.