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Open Access Publications from the University of California

Helly numbers of algebraic subsets of Rdand an extension of doignon's theorem

  • Author(s): De Loera, JA
  • La Haye, RN
  • Oliveros, D
  • Roldán-Pensado, E
  • et al.

© 2017 by Walter de Gruyter Berlin/Boston 2017. We study S-convex sets, which are the geometric objects obtained as the intersection of the usual convex sets in Rdwith 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: low-dimensional situations, sets that have some algebraic structure, in particular when S is an arbitrary subgroup of Rdor 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.

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