Example 12 Find the smallest positive integer $n$, such that in a simple graph with 10 vertices and $n$ edges, there always exists a monochromatic triangle or a monochromatic quadrilateral (i.e., a graph containing 4 edges).
(1996 Chinese National Training Team Test Question)
In Figure 7-13, there are 10 vertices and 30 edges, with 15 red edges (solid lines) and 15 blue edges (dashed lines). The graph contains neither monochromatic triangles nor monochromatic quadrilaterals, so the smallest positive integer \( n \geqslant 31 \).

Below is the proof that when \( n=31 \), the graph must contain a monochromatic triangle or a monochromatic quadrilateral. By the pigeonhole principle, the graph must have \(\left[\frac{31-1}{2}\right]+1=16\) edges of the same color. Therefore, it is only necessary to prove that in a simple graph with 10 vertices and 16 edges, there must be a triangle or a quadrilateral.

Since 16 edges have 32 endpoints, there must be a vertex that is incident to at most 3 edges. Without loss of generality, assume \( A_{10} \) is incident to at most 3 edges. Thus, the other 9 vertices must be incident to at least \( 16-3=13 \) edges, and 13 edges have 26 endpoints (any extra edges can be deleted without affecting the subsequent proof). Therefore, there must be a vertex that is incident to at most 2 edges. Without loss of generality, assume \( A_{9} \) is incident to 2 edges. Thus, the subgraph with 8 vertices \(\left\{A_{1}, A_{2}, \cdots, A_{8}\right\}\) has 11 edges. Since 11 edges have 22 endpoints, there must be a vertex that is incident to at most 2 edges. Without loss of generality, assume \( A_{8} \) is incident to 2 edges. Thus, the subgraph with 7 vertices \(\left\{A_{1}, \cdots, A_{7}\right\}\) has 9 edges. Similarly, the subgraph with 6 vertices \(\left\{A_{1}, \cdots, A_{6}\right\}\) must have 7 edges. Since \(7>6-1\), this simple graph is not a tree and must contain a cycle. If the cycle with the most edges has 6 edges, then the 7th edge, no matter how it is connected, will always form a triangle or a quadrilateral with two of its endpoints; if the cycle with the most edges has 5 edges, then the other two edges either share a common endpoint outside the cycle or at least one of them connects two points on the cycle, in either case forming a triangle or a quadrilateral; if the cycle with the most edges has at most 4 edges, the conclusion is obviously true.
In summary, the smallest positive integer \( n=31 \).