Let there be $m$ points in the Euclidean space $\mathbb R^n$. Randomly choose $k$ distinct pairs of these $m$ points, and assign a random (positive) value for the Euclidean distance between each of these $k$ pairs.

Determine the maximum value of $k$ as a function of $n$ and $m$ such that, for any random choice of $k$ distinct pairs of points and the Euclidean distances between the points, either

- There exists some configuration of $m$ points satisfying all the distance relationships, OR
- There exists a triplet of points for which all three pairwise distances are defined, and these three distances do not satisfy the triangle inequality.

The question above is similar to this one Reconstructing an Euclidean point cloud from their pairwise distances (and others like it) but I believe the math involved is different.