Helly numbers of algebraic subsets of ℝd and an extension of Doignon's Theorem.

We study S-convex sets, which are the geometric objects obtained as the intersection of the usual convex sets in ℝd with a proper subset S ⊂ ℝd, and contribute new results about their S-Helly numbers. We extend prior work for S = ℝd, ℤd, and ℤd-k × ℝk, and give some sharp bounds for several new cases: lowdimensional situations, sets that have some algebraic structure, in particularwhen S is an arbitrary subgroup of ℝd or when S is the difference between a lattice and some of its sublattices. By abstracting the ingredients of Lovász method we obtain colorful versions of many monochromatic Helly-type results, including several colorful versions of our own results. [ABSTRACT FROM AUTHOR]