Extensions of discrete Helly theorems for boxes

We prove extensions of Halman's discrete Helly theorem for axis-parallel boxes in Rᵈ. Halman's theorem says that, given a set S in Rᵈ, if F is a finite family of axis-parallel boxes such that the intersection of any 2d contains a point of S, then the intersection of F contains a point of S. We prove colorful, fractional, and quantitative versions of Halman's theorem. For the fractional versions, it is enough to check that many (d+1) -tuples of the family contain points of S. Among the colorful versions we include variants where the coloring condition is replaced by an arbitrary matroid. Our results generalize beyond axis-parallel boxes to H-convex sets.