Problem 1921

Let $a_n$ be the maximum possible number of sections of a set of $n$ points in the plane.
Let $S$ be a set of $n+1$ points in the plane and let us consider one of these points, call it $T$, on the convex hull of that set. Note that each section of $S$ restricts to a section of the set $S \setminus \{T\}$. Let $\{A, B\}$ and $\{C, D\}$ be two different sections of the set $S$ that restrict to the same section of the set $S \setminus \{T\}$. This means that $\{A \setminus \{T\}, B \setminus \{T\}\} = \{C \setminus \{T\}, D \setminus \{T\}\}$. Furthermore, it means, without loss of generality, that the sets $A$ and $C$ are the same and the sets $B$ and $D$ are the same, up to the point $T$. Hence, for the section $\{A \setminus \{T\}, B \setminus \{T\}\}$ of the set $S \setminus \{T\}$ we can take a line passing through the point $T$ such that all the points of the set $A \setminus \{T\}$ are on one side of the line, while all the points of the set $B \setminus \{T\}$ are on the other.
Let us consider all the lines passing through the point $T$ and none of the points of the set $S \setminus \{T\}$. Two such lines could give two different sections of the set $S \setminus \{T\}$ only if there is at least one point of the set $S \setminus \{T\}$ between them. Hence, the number of different sections of the set $S \setminus \{T\}$ corresponding to a line passing through $T$ is at most $n$, i.e. as many as there are points in the set $S \setminus \{T\}$.
Finally, we note that
$$a_{n+1} \le n + a_n \le n + (n-1) + a_{n-1} \le \dots \le n + (n-1) + \dots + 2 + 1 + a_1 = \binom{n+1}{2} + 1.$$
So, the set of $n$ points in the plane can have at most $\binom{n}{2} + 1$ sections. That number is achievable, e.g. in the case of a regular $n$-gon.