3. Given $P$ as a regular $n$-sided polygon inscribed in a unit circle, $M(n)$ as the maximum number of points within $P$ or on its boundary such that the distance between any two points is not less than 1. Then the set of positive integers $n$ that satisfy $M(n)=n+1$ is


3. $\{3,4,5,6\}$.

As shown. When $n=3,4,5,6$, the requirements are met.
When $n \geqslant 7$, the center $O$ and $n$ vertices $A_{1}, A_{2}, \cdots, A_{n}$ divide $P$ into $n$ triangles. By the pigeonhole principle, there must be a triangle with 2 points inside (as shown in Figure 2), suppose it is in $\triangle O A_{1} A_{2}$. Knowing that $A_{1} A_{2}<1$, the two points must be on $O$. Thus, the remaining $n$ points are all at vertex $A_{i}$, and since $A_{i} A_{i+1}<1$, the rest follows.