A lower bound on the crossing number of uniform hypergraphs

In this paper, we consider the embedding of a complete dd-uniform geometric hypergraph with nn vertices in general position in RdRd, where each hyperedge is represented as a (d−1)(d−1)-simplex, and a pair of hyperedges is defined to cross if they are vertex-disjoint and contain a common point in the relative interiors 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.