Would this be bounded

Question

Let $a_{m}$ be supremum of the minimum of the angle between the line segments between any $m$ points, in which the supremum is taken over all configurations of $m$ points. Is $\sqrt{m}a_{m}$ bounded as $m\rightarrow\infty$ ?

Answer 1

In $\mathbb{R}^2$ it seems unlikely that $a_m\sqrt{m}$ is bounded.

It is known that there is some $c>0$ such that the number of distinct angles in $[0,\pi)$ determined by $3$-point subsets of any set of $m$ non-collinear points is at least $cm$. It is conjectured that the minimum number of angles determined by such a point set is in fact $m-2$.

The number of distinct directions determined by $m$ non-collinear points in the plane is $m$ or $m-1$ depending on whether $m$ is even or odd.

See section 6.2 of Research problems in discrete geometry.