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Let $X$ be a convex curve in the plane (say, the unit circle), and let$\mathcal S$ be a family of planar convex bodies, such that every two of themmeet at a point of $X$. Then $\mathcal S$ has a transversal $N\subset\mathbbR^2$ of size at most $1.75\cdot 10^9$.  Suppose instead that $\mathcal S$ only satisfies the following"$(p,2)$-condition": Among every $p$ elements of $\mathcal S$ there are twothat meet at a common point of $X$. Then $\mathcal S$ has a transversal of size$O(p^8)$. For comparison, the best known bound for the Hadwiger--Debrunner $(p,q)$-problem in the plane, with $q=3$, is $O(p^6)$.  Our result generalizes appropriately for $\mathbb R^d$ if $X\subset \mathbbR^d$ is, for example, the moment curve.

A variant of the Hadwiger-Debrunner (p,q)-problem in the plane