, 2026

Let $\mathcal{P}$ be a set of ten points in the plane. A subset $\mathcal{Q}\subseteq \mathcal{P}$ is said to be isolable if there exists a closed disk of positive integer radius that contains all points of $\mathcal{Q}$ but none of the points of $\mathcal{P}\setminus\mathcal{Q}$. Prove that at least one third of the five-element subsets of $\mathcal{P}$ are not isolable.