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 (Formula presented.) of size at most 1.75×10⁹. 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(p⁸). For comparison, the best known bound for the Hadwiger–Debrunner (p, q)-problem in the plane, with q=3, is O(p⁶). Our result generalizes appropriately for (Formula presented.) if (Formula presented.) is, for example, the moment curve.