22. Find the smallest positive integer $n$ with the following property: whenever any 5 vertices of a regular $n$-gon $S$ are colored red, there is always a line of symmetry $l$ of $S$ such that the reflection of each red point across $l$ is not a red point.
22. For a regular 13-sided polygon, when the vertices $A_{1}, A_{2}, A_{4}, A_{6}, A_{7}$ are painted red, none of the 13 axes of symmetry satisfy the problem's requirements. For regular 12, 11, and 10-sided polygons, when the above 5 points are red, the conclusion of the problem also does not hold. When the number of sides is no more than 9, the conclusion is obviously not valid, so the smallest integer $n \geqslant 14$. For a regular 14-sided polygon, $A_{1} A_{2} \cdots A_{14}$ has 7 axes of symmetry that do not pass through the vertices but bisect a pair of opposite sides. We classify $A_{i}$ as odd or even based on the parity of the index $i$. Clearly, when one of the above 7 axes of symmetry is the axis of symmetry, the parity of each pair of symmetric points is different. Let there be $m$ odd vertices among the 5 red vertices, $0 \leqslant m \leqslant 5$, then there are $5-m$ even vertices. The number of lines connecting red odd vertices and red even vertices is $m(5-m) \leqslant \left[\frac{m+(5-m)}{2}\right]^{2}=6 \frac{1}{4}$. Since $m(5-m)$ is an integer, $m(5-m) \leqslant 6$. These lines have at most 6 perpendicular bisectors, so at least one of the above 7 axes of symmetry does not bisect any of these 6 lines perpendicularly. This axis is the one we seek. In summary, the smallest value of the number of sides $n$ is 14.