7. Find the smallest natural number $n$, such that in any two-coloring of $K_{n}$, there always exist two monochromatic triangles, one of which shares exactly one edge with the other.

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7. Find the smallest natural number $n$, such that in any two-coloring of $K_{n}$, there always exist two monochromatic triangles, one of which shares exactly one edge with the other.
7. Solution: When $n \geqslant 10$, take one of the $K_{10}$, it has at least 20 monochromatic triangles, with a total of $20 \times 3=60$ edges, and $60>\mathrm{C}_{10}^{2}$, so there exist two monochromatic triangles that share exactly one edge.
When $n=9$, as shown in the figure.
In the figure, the connected edges are blue edges; the rest of the unconnected edges are red edges, i.e., the edges in the complement graph of the figure (please draw it yourself).

It can be seen that there are no two monochromatic triangles sharing an edge in both the graph and its complement graph.
Therefore, $n_{\min }=10$.