(12) (22 points) Suppose a set of planar points $S$ has the properties: (i) no three points are collinear; (ii) the distances between any two points are all different. For two points $A$ and $B$ in $S$, if there exists $C \in S$ such that $|A C|<|A B|<|B C|$, then $A B$ is called a middle edge of $S$. For three points $A$, $B$, and $C$ in $S$, if $A B$, $A C$, and $B C$ are all middle edges of $S$, then $\triangle A B C$ is called a middle edge triangle of $S$. Find the smallest $n$ such that any $n$-element planar point set $S$ with properties (i) and (ii) must contain a middle edge triangle.
(12) (1) Color all the middle edges of $S$ red, and all other edges blue. When $n \geqslant 6$, according to Ramsey's theorem, there must exist a monochromatic triangle, which must have a middle edge, and thus must be a middle edge triangle.
(2) There exists a 5-element point set with properties (i) and (ii), but it does not have a middle edge triangle, for example: