4. 7 (CMO17) For four distinct points $P_{1}, P_{2}, P_{3}, P_{4}$ in the plane, find the minimum value of the ratio $\frac{\sum_{1 \leqslant i<j \leqslant 4} P_{i} P_{j}}{\min _{1 \leqslant j \leqslant 4} P_{i} P_{j}}$.
4. 7 First, prove the following lemma: In $\triangle ABC$, if $AB \geqslant m, AC \geqslant m, \angle BAC = \alpha$, then $BC \geqslant 2m \sin \alpha / 2$. This can be proven using the Law of Sines.

Let the numerator of the original expression be $k$, and the denominator be $m$. Estimate the value of $k: m$ by categorizing. For example, if the convex hull of the four points is a triangle, assume without loss of generality that point $P_{4}$ is inside $\triangle P_{1} P_{2} P_{3}$, and assume $\angle P_{1} P_{4} P_{2} \geqslant 120^{\circ}$.

By the lemma, $P_{1} P_{2} \geqslant 2m \sin \frac{\angle P_{1} P_{4} P_{2}}{2} \geqslant 2m \sin 60^{\circ} = \sqrt{3} m$, thus, $\frac{k}{m} \geqslant 5 + \sqrt{3}$.
For other cases (three of the four points are collinear; the convex hull is a quadrilateral), the discussion is similar. The minimum value sought is $\sqrt{3} + 5$.