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

Let X be a convex curve in the plane (say, the unit circle), and let 𝒮 be a family of planar convex bodies such that every two of them meet at a point of X . Then 𝒮 has a transversal N⊂ℝ^2 of size at most 1.75× 10^9 . Suppose instead that 𝒮 only satisfies the following “( p , 2)-condition”: Among every p elements of 𝒮 , there are two that meet at a common point of X . Then 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 ℝ^d if X⊂ℝ^d is, for example, the moment curve.