The (2,2) and (4,3) properties in families of fat sets in the plane

A family of sets satisfies the $(p,q)$ property if among every $p$ members of it some $q$ intersect. Given a number $0<r\le 1$, a set $S\subset \mathbb{R}^2$ is called $r$-fat if there exists a point $c\in S$ such that $B(c,r) \subseteq S\subseteq B(c,1)$, where $B(c,r)\subset \mathbb{R}^2$ is a disk of radius $r$ with center-point $c$. We prove constant upper bounds $C=C(r)$ on the piercing numbers in families of $r$-fat sets in $\mathbb{R}^2$ that satisfy the $(2,2)$ or the $(4,3)$ properties. This extends results by Danzer and Karasev on the piercing numbers in intersecting families of disks in the plane, as well as a result by Kyn\v{c}l and Tancer on the piercing numbers in families of units disks in the plane satisfying the $(4,3)$ property.