Consider a 2-dimensional Borsuk's theorem:

Every bounded set $S$ in the plane can be partitioned into three parts with diameter smaller than the diameter of $S$.

I wonder if there are any results on variations of this problem which consider the following questions:

partitioning a

**finite**plane set $S$ into parts with the minimum of the distances between its different points, larger than the minimum of the distances between different points of $S$;partitioning a

**finite**plane set $S$ into parts with minimum of the distances between its different points, larger than the minimum of the distances between different points of $S$, and/or with diameter, smaller than the diameter of $S$;maybe other questions, connected with extremal characteristics of

**finite**plane sets, other than diameter?