Set S is overdetermined if |S| ≥ 2 and $a$ polynomial of degree ≤ |S|-2 passes through it. Max overdetermined subsets for non-overdetermined $n$ points?

Step 1 (Overdetermined Subset Definition): A set $S$ of points is overdetermined if there exists a polynomial of degree $\le |S|-2$ passing through all points in $S$. In 2D space, this means that for any $|S|\ge 2$, the points lie on a algebraic curve of lower degree. Specifically, for $|S|=3$, overdetermined means the three points are collinear.

Step 2 (The Collinearity Constraint): We are given that the set of $n$ points does not contain any overdetermined subset of size $\ge 3$. This means no three points are collinear. Under this constraint, any overdetermined subset must have size exactly 2 (since any 2 points are trivially collinear, i.e., lie on a line of degree 0, which is $\le 2-2=0$).

Step 3 (Counting Subsets): By setting up the induction on $n$, the maximum number of overdetermined subsets is achieved when $n-1$ points lie on a single line (which is forbidden by the size $\ge 3$ rule, so we must analyze the boundary). The maximum number of such overdetermined subsets is exactly $2^{n-1}-n$. $■$