Bounding Radon Number via Betti Numbers

Author Zuzana Patáková

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Zuzana Patáková
  • Computer Science Institute, Charles University, Prague, Czech Republic
  • IST Austria, Klosterneuburg, Austria


First and foremost, I am very grateful to Pavel Paták for numerous discussions, helpful suggestions and a proofreading. Many thanks to Xavier Goaoc for his feedback and comments, which have been very helpful in improving the overall presentation. I also thank Endre Makai for pointers to relevant literature, especially to the book [Soltan, 1984]. Finally, many thanks to Natan Rubin for several discussions at the very beginning of the project.

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Zuzana Patáková. Bounding Radon Number via Betti Numbers. In 36th International Symposium on Computational Geometry (SoCG 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 164, pp. 61:1-61:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)


We prove general topological Radon-type theorems for sets in ℝ^d, smooth real manifolds or finite dimensional simplicial complexes. Combined with a recent result of Holmsen and Lee, it gives fractional Helly theorem, and consequently the existence of weak ε-nets as well as a (p,q)-theorem. More precisely: Let X be either ℝ^d, smooth real d-manifold, or a finite d-dimensional simplicial complex. Then if F is a finite, intersection-closed family of sets in X such that the ith reduced Betti number (with ℤ₂ coefficients) of any set in F is at most b for every non-negative integer i less or equal to k, then the Radon number of F is bounded in terms of b and X. Here k is the smallest integer larger or equal to d/2 - 1 if X = ℝ^d; k=d-1 if X is a smooth real d-manifold and not a surface, k=0 if X is a surface and k=d if X is a d-dimensional simplicial complex. Using the recent result of the author and Kalai, we manage to prove the following optimal bound on fractional Helly number for families of open sets in a surface: Let F be a finite family of open sets in a surface S such that the intersection of any subfamily of F is either empty, or path-connected. Then the fractional Helly number of F is at most three. This also settles a conjecture of Holmsen, Kim, and Lee about an existence of a (p,q)-theorem for open subsets of a surface.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Geometric topology
  • Theory of computation → Computational geometry
  • Radon number
  • topological complexity
  • constrained chain maps
  • fractional Helly theorem
  • convexity spaces


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