A Variant of the Hadwiger–Debrunner (p, q)-Problem in the Plane

Abstract

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 $1.75\times 10^9$. 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.