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In this question, a closed triangle on a plane is a set of all points in its area and on its boundary, while an open triangle excludes its boundary.

Now, the problems: Let $T$ be an equilateral triangle with sides of $1$ on a plane, $S = \{ A_1, ..., A_s \} \subset T$ be a set of some distinct points.

The general problem: Determine $$\max_{S \subset T} \left [ \min_{1 \le i < j \le s} d(A_i,A_j) \right ],$$ if it exists; otherwise, determine $$\sup_{S \subset T} \left [ \min_{1 \le i < j \le s} d(A_i,A_j) \right ].$$ Assume that:

$i) \; T \text{ is closed}, \; s = n^2+1 \; (n \ge 4)$.

$ii) \; T \text{ is closed}, \; s = n^2 \; (n \ge 3)$.

$iii) \; T \text{ is open}, \; s = n^2 \; (n \ge 3)$.

$(n \in \mathbb{Z^{+}})$


This problem is posed by myself out of curiosity when I considered $ii, \; iii$ in the case of $n = 3$. I haven't solved even this particular case so far. Therefore, we can start with this case, which is called The specific problem from now on. Also, FYI, the common situation (?) of the problems is originated from another problem which can be easily proved by the Pigeonhole Principle.

Thanks in advance!!!

P.S: I don't know exactly what to title and what to tag. So, if I had it wrong, please help me fix it, and thank you for that!

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