It is known by the pigeon-hole principle that:

- If we select $5$ points within an equilateral triangle with side $1$, there must be at least two whose distance apart is less than or equal to $1/2$.
- And if we select $10$ points, there must be at least two whose distance apart is $\leq 1/3$.
- Generally, if we select $n^2+1$ points, there must be at least two with distance $\leq 1/n$.

But $n^2+1$ seems not to be a tight bound. My question is:

To determine a

minimuminteger $m(n)$ such that if we select $m(n)$ points within an equilateral triangle with side $1$, there must be at least two points having distance $\leq 1/n$.

Equivalently,

To determine the

maximuminteger $m(n)$ satisfying that there exists a configuration of $m(n)-1$ points within an equilateral triangle with side $1$ such that the minimal distance among these points is greater than $1/n$.

It is clear that $m(1)=2$ and $m(2)=5$, both matching $n^2+1$. But a roughly pencil-and-paper work shows that $m(3)$ is not $10$ anymore.

Note: points can be located on the three sides of the triangle.