How can we find the infimum of the following set $$\left\{\frac{\max_{i\neq j}d(A_i,A_j)}{\min_{i\neq j}d(A_i,A_j)}:A_1,A_2,\ldots,A_6 \in \mathbb R^2, A_i\neq A_j (\forall i\neq j)\right\}.$$
This comes from a meme, but I thought it might be interesting to know what this infimum will be. But I did not make much progress on it (except for finding the ratio for some special cases).