Example 2. Find the smallest positive integer $n$, such that in any 9-vertex and $n$-edge graph with edges colored in two colors, there must exist a monochromatic triangle.

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Since $K_{2}$ can have one edge removed and be colored with 0 colors to ensure it contains no monochromatic triangles, and $K_{\mathrm{s}}$ can be colored with 2 colors to ensure it contains no monochromatic triangles. $K$, can be divided into 4 $K_{2}$'s and 1 $K_{1}$ with no common vertices, so in $K$, 4 edges can be removed and then colored with $2+0=2$ colors to ensure it contains no monochromatic triangles. The specific coloring method is shown in the figure. Therefore, to ensure that $K$, after removing $m$ edges, any 2-coloring contains a monochromatic triangle, it must be that $m \leqslant 3$.

When 3 edges are removed from $K$, it must contain a complete graph $K_{6}$ (each removed edge removes one endpoint, leaving 6 vertices to form $K_{6}$). Therefore, in $K_{9}$, after arbitrarily removing 3 edges, any 2-coloring must contain a monochromatic triangle. Thus, the smallest positive integer
$$
n=C_{9}^{2}-3=33 .
$$