  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 them
meet at a point of $X$. Then $\mathcal S$ has a transversal $N\subset\mathbb
R^2$ 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 two
that 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 $X\subset \mathbb
R^d$ is, for example, the moment curve.
