Question

It is known by the pigeon-hole principle that:

1. 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 $1/2$.
2. And if we select $10$ points, there must be at least two whose distance apart is $\leq 1/3$.
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 minimum integer $m(n)$ 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 maximum integer $m(n)$ 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.

Answer 1

This is related to problems of packing circles into an equilateral triangle, and covering an equilateral triangle by circles. Some data on these problems is available at http://www2.stetson.edu/~efriedma/cirintri/ and http://www2.stetson.edu/~efriedma/circovtri/

Since the Heilbronn triangle problem has been mentioned, that has data at http://www2.stetson.edu/~efriedma/heiltri/