6. In the plane, the partition of a finite set of points refers to dividing the point set into two non-intersecting subsets $A$ and $B$, such that there exists a line that does not pass through any point in the set and all points in subset $A$ are on one side of the line, and all points in subset $B$ are on the other side. Find the maximum number of partitions of a set of $n$ points in the plane.
6. Let the maximum number of partitions of a set of $n$ points in the plane be $a_{n}$.

For a set $S$ of $n+1$ points in the plane, consider one of the points in the convex hull of these points, called $T$. On the convex hull of this set, note that each partition of $S$ restricts a partition of $S \backslash\{T\}$.

Let $\{A, B\}$ and $\{C, D\}$ be two different partitions of $S$ that restrict the same partition of $S \backslash\{T\}$. Typically, $A$ is the same as $C$, and $B$ is the same as $D$, depending on the point $T$.

Thus, for the part $\{A \backslash\{T\}, B \backslash\{T\}\}$ of $S \backslash\{T\}$, a line passing through point $T$ can be chosen such that all points in $A \backslash\{T\}$ are on one side of the line, and all points in $B \backslash\{T\}$ are on the other side.

Now consider all lines passing through point $T$ and not containing any points from $S \backslash\{T\}$.

These two lines can divide $S \backslash\{T\}$ into two different partitions only if there is at least one point in $S \backslash\{T\}$ between them. Therefore, the number of different partitions of $S \backslash\{T\}$ is equivalent to the maximum number of lines passing through $T$ being $n$, which is the number of points in $S \backslash\{T\}$.
Finally, note that,
$$
\begin{array}{l}
a_{n+1} \leqslant n+a_{n} \leqslant n(n+1)+a_{n-1} \leqslant \cdots \\
\leqslant n+(n-1)+\cdots+2+1+a_{1} \\
=\mathrm{C}_{n+1}^{2}+1 .
\end{array}
$$

Thus, the maximum number of partitions of a set of $n$ points in the plane is $\mathrm{C}_{n}^{2}+1$, as in a regular $n$-gon.