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 $1.75\cdot 10^9$. 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.

10 pages, 1 figure