$4 \cdot 37$ Find the smallest positive integer $n$, such that when $K_{n}$ is colored in two colors in any way, there always exist two monochromatic triangles, each of which shares exactly 1 edge.
[Solution] In the $K_{9}$ on the right, for clarity, we have drawn the blue and red lines in two separate diagrams. It is easy to see that there are 6 blue and 6 red, a total of 12 monochromatic triangles in the graph. The 36 edges in the graph are exactly one edge of a monochromatic triangle each, and no two monochromatic triangles share a common edge. Therefore, the smallest positive integer $n \geqslant 10$.

In a two-colored $K_{10}$, using the method of calculating the sum of bichromatic angles and bichromatic triangles, it is not difficult to calculate that there are at least 20 monochromatic triangles. Each triangle has 3 edges, so these monochromatic triangles have at least 60 edges in total. However, the number of edges in $K_{10}$ is $C_{10}^{2}=45$, so there must be two monochromatic triangles that share exactly 1 common edge.
In summary, the smallest positive integer $n=10$.