Anshu, AnuragShannigrahi, Saswata2024-02-092016-08Anshu, Anurag, Saswata Shannigrahi. "A lower bound on the crossing number of uniform hypergraphs." Discrete Applied Mathematics 209 (2016): 11-15. DOI: 10.1016/j.dam.2015.10.0090166-218Xhttps://nrs.harvard.edu/URN-3:HUL.INSTREPOS:37377736In this paper, we consider the embedding of a complete d-uniform geometric hypergraph with n vertices in general position in ℝd, where each hyperedge is represented as a (d−1)-simplex, and a pair of hyperedges is defined to cross if they are vertex-disjoint and contains a common point in the relative interior of the simplices corresponding to them. As a corollary of the Van Kampen-Flores Theorem, it can be seen that such a hypergraph contains Ω(2dd√) (n2d) crossing pairs of hyperedges. Using Gale Transform and Ham Sandwich Theorem, we improve this lower bound to Ω(2dlogdd√) (n2d).en-USApplied MathematicsDiscrete Mathematics and CombinatoricsJournal Article2024-02-0910.1016/j.dam.2015.10.009