New Quantitative Bounds for the $(p,q)$-Theorem for Unions of Convex Sets

Chaya Keller, Shakhar Smorodinsky

A set in ${\mathbb{R}}^d$ is $s$-convex if it is the union of at most $s$ convex sets. A family $F$ satisfies the $(p,q)$ property if among any $p$ sets in $F$, some $q$ intersect. L…