MathNet 2023-part-04 — 191 problems (EN–VI) Question 175

Fix a positive integer $n$. Consider an $n$-point set $S$ in the plane. An *eligible* set is a non-empty set of the form $S \cap D$, where $D$ is a closed disc in the plane. In terms of $n$, determine the smallest possible number of eligible subsets $S$ may contain.
Cristian Săvescu