It is evident true that there are at most 4 points are equal-distance in three dimention Euclidean Space.

What about other metric space? This ideal was first generated by my student Yilin Liu, and some results were obtained by us together.

- In a discrete metric, $X$ is any non-empty set.

$$ \rho(x,y) = \left \lbrace \begin{array}{ll} 1 & x \neq y \\ 0 & x=y \end{array} \right. $$

In this metric space,any subset of $X$ is equal-distance. - Suppose that $X$ if we define

$$ \rho(A,B) = |x_1 - x_2| + |y_1 - y_2| + |z_1 - z_2| $$,

where $A(x_1,y_1,z_1),B(x_2,y_2,z_2)$.

we can find six points satisfy every two of them have some distance,for instance,$(\pm 1,0,0)\;(0,\pm 1,0)\;(0,0,\pm 1)$. - Suppose that $X$ if we define

$$ \rho(A,B) = \sqrt{(x_1 - x_2)^2 + (y_1 - y_2)^2 }+ |z_1 - z_2| $$,

where $A(x_1,y_1,z_1),B(x_2,y_2,z_2)$.

It is hard to find more that 4 point which could feed our demand. Fortunately with a skill construction,we finally get five equal-distance point set as follow

$$ (1,0,1); \;(-1,0,-1); \;(-\sqrt{2},0,4-\sqrt{2}); \;(-\frac{3\sqrt{2}}{2},2,\frac{\sqrt{2}}{2}); \;(-\frac{3\sqrt{2}}{2},-2,\frac{\sqrt{2}}{2}) $$

What more, it remains to be considered in n-dimention space,or any metric and topology space.