Example 4.3.8 Let $s$ be the set of points inside and on the boundary of a regular hexagon with side length 1. Find the smallest $r$ such that there exists a way to color the points in $s$ with three colors, so that the distance between any two points of the same color is less than $r$.
Solution: When $r=\frac{3}{2}$, it is feasible, as shown in figures I, II, III, using two colors to dye, satisfying the requirements.

Now prove that for any $r<\frac{3}{2}$, there exists $\varepsilon>0$, such that $r+\varepsilon<r, A^{\prime} C>r, A^{\prime}$ and $B, C$ are different colors, $A^{\prime}$ must be the same color as $A$. For any point $A$, points within a distance of $\varepsilon$ from $A$ are all the same color as $A$, thus the entire circumference is the same color, leading to a contradiction.