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$ K $-Sets and Rectilinear Crossings in Complete Uniform Hypergraphs k-Sets and Rectilinear Crossings in Complete Uniform Hypergraphs Rahul Gangopadhyay1 and Saswata Shannigrahi2 1 IIIT Delhi, India [email protected] 2 Saint Petersburg State University, Russia [email protected] Abstract. In this paper, we study the d-dimensional rectilinear drawings d of the complete d-uniform hypergraph K2d. Anshu et al. [Computational Geometry: Theory and Applications, 2017] used Gale transform and Ham- Sandwich theorem to prove that there exist Ω 2d crossing pairs of hy- d peredges in such a drawing of K2d. We improve this lower bound by showing that there exist Ω 2d√d crossing pairs of hyperedges in a d- d dimensional rectilinear drawing of K2d. We also prove the following re- sults. 1. There are Ω 2dd3/2 crossing pairs of hyperedges in a d-dimensional d rectilinear drawing of K2d when its 2d vertices are either not in convex position in Rd or form the vertices of a d-dimensional convex polytope that is t-neighborly but not (t + 1)-neighborly for some constant t 1 ≥ independent of d. 2. There are Ω 2dd5/2 crossing pairs of hyperedges in a d-dimensional d rectilinear drawing of K2d when its 2d vertices form the vertices of a d- dimensional convex polytope that is ( d/2 t )-neighborly for some ⌊ ⌋ − ′ constant t 0 independent of d. ′ ≥ Keywords: Rectilinear Crossing Number · Neighborly Polytope · Gale · · · · arXiv:1806.02574v5 [math.CO] 28 Oct 2019 Transform Affine Gale Diagram Balanced Line j-Facet k-Set 1 Introduction d For a sufficiently large positive integer d, let Kn denote the complete d-uniform d hypergraph with n 2d vertices. A d-dimensional rectilinear drawing of Kn is de- ≥ Rd d fined as an embedding of it in such that the vertices of Kn are points in gen- eral position in Rd and each hyperedge is drawn as the (d 1)-simplex formed − by the d corresponding vertices [4]. Note that a set of points in Rd is said to be 2 R. Gangopadhyay et al. in general position if no d + 1 of them lie on a (d 1)-dimensional hyperplane. − For u and v in the range 0 u, v d 1, a u-simplex spanned by a set of u + 1 ≤ ≤ − points and a v-simplex spanned by a set of v + 1 points (when these u + v + 2 points are in general position in Rd) are said to be crossing if they do not have common vertices (0-faces) and contain a common point in their relative interi- ors [5]. As a result, a pair of hyperedges in a d-dimensional rectilinear drawing d of Kn are said to be crossing if they do not have a common vertex and contain a common point in their relative interiors [3,4,5]. The d-dimensional rectilinear d d crossing number of Kn, denoted by crd(Kn), is defined as the minimum number of crossing pairs of hyperedges among all d-dimensional rectilinear drawings d of Kn [3,4]. Since the crossing pairs of hyperedges formed by a set containing 2d vertices d of Kn are distinct from the crossing pairs of hyperedges formed by another set n of 2d vertices, it can be observed that cr (Kd) cr (Kd ) [4]. The best- d n ≥ d 2d 2d d Ω d known lower bound on crd(K2d) is (2 ) [3]. Anshu et al. [3] also studied a d d-dimensional rectilinear drawing of K2d where all its vertices are placed on the d-dimensional moment curve γ = (a, a2,..., ad) : a R and proved that the { ∈ } number of crossing pairs of hyperedges is Θ 4d/√d in this drawing. d As described above, an improvement of the lower bound on crd(K2d) im- d proves the lower bound on crd(Kn). Let us denote the points corresponding to d the set of vertices in a d-dimensional rectilinear drawing of the hypergraph K2d by V = v , v ,..., v . The points in V are said to be in convex position if there { 1 2 2d} does not exist any point v V (for some i in the range 1 i 2d) such that v i ∈ ≤ ≤ i can be expressed as a convex combination of the points in V vi , and such a d \ { } drawing of K2d is called a d-dimensional convex drawing of it [3]. d Note that the convex hull of the vertices of K2d in a d-dimensional convex drawing of it forms a d-dimensional convex polytope with its vertices in general position. For any t 1, a d-dimensional t-neighborly polytope is a d-dimensional ≥ convex polytope in which each subset of its vertex set having less than or equal to t vertices forms a face [7, Page 122]. Such a polytope is said to be neighborly if it is d/2 -neighborly. Note that any d-dimensional convex polytope can be at ⌊ ⌋ most d/2 -neighborly unless it is a d-simplex which is d-neighborly [7, Page ⌊ ⌋ 123]. The d-dimensional cyclic polytope is a special kind of neighborly polytope where all its vertices are placed on the d-dimensional moment curve [14, Page 15]. Using these definitions and notations, let us describe our contributions in this paper. d In Section 4, we improve the lower bound on crd(K2d). In particular, we prove the following theorem which implies Corollary 1. d Ω d√ Theorem 1. crd(K2d) = (2 d). n Corollary 1. cr (Kd) = Ω(2d√d) . d n 2d k-Sets and Rectilinear Crossings in Complete Uniform Hypergraphs 3 We then prove the following theorems to obtain lower bounds on the number of crossing pairs of hyperedges in some special types of d-dimensional rectilinear d drawings of K2d. Theorem 2. The number of crossing pairs of hyperedges in a d-dimensional rectilinear d Ω d 3/2 d drawing of K2d is (2 d ) if the vertices of K2d are not in convex position. Theorem 3. For any constant t 1 independent of d, the number of crossing pairs of ≥ d Ω d 3/2 hyperedges in a d-dimensional rectilinear drawing of K2d is (2 d ) if the vertices d of K2d are placed as the vertices of a d-dimensional t-neighborly polytope that is not (t + 1)-neighborly. Theorem 4. For any constant t′ 0 independent of d, the number of crossing pairs of ≥ d Ω d 5/2 hyperedges in a d-dimensional rectilinear drawing of K2d is (2 d ) if the vertices of Kd are placed as the vertices of a d-dimensional ( d/2 t )-neighborly polytope. 2d ⌊ ⌋ − ′ Note that the t′ = 0 case in Theorem 4 corresponds to a neighborly poly- tope. As mentioned above, the number of crossing pairs of hyperedges in a d Θ d √ d-dimensional rectilinear drawing of K2d is known to be (4 / d) when such a polytope is cyclic. Techniques Used: We use the properties of Gale transform, affine Gale dia- gram, k-sets and balanced lines to prove Theorems 1, 2, 3 and 4. We discuss these properties in detail in Sections 2 and 3. In addition, a few other results used in these proofs are mentioned below. Let us first state the Ham-Sandwich theorem and the Caratheodory’s theorem that are used in the proofs of Theo- rems 1 and 2, respectively. Ham-Sandwich Theorem: [9,13] There exists a (d 1)-hyperplane h which si- − Rd multaneously bisects d finite point sets P1, P2,..., Pd in , such that each of the open half-spaces created by h contains at most P /2 points of P for each i in ⌊| i| ⌋ i the range 1 i d. ≤ ≤ Carath´eodory’s Theorem: [9,12] Let X Rd. Then, each point in the convex ⊆ hull Conv(X) of X can be expressed as a convex combination of at most d + 1 points in X. In the following, we state the Proper Separation theorem that is used in the proof of Lemma 1. Two non-empty convex sets are said to be properly sepa- rated in Rd if they lie in the opposite closed half-spaces created by a (d 1)- − dimensional hyperplane and both of them are not contained in the hyperplane. Proper Separation Theorem: [8, Page 148] Two non-empty convex sets can be properly separated in Rd if and only if their relative interiors are disjoint. 4 R. Gangopadhyay et al. The proof of the following lemma, which is used in the proofs of all four theo- rems, is the same proof mentioned in [3] for the special case u = v = d 1. For − the sake of completeness, we repeat its proof in full generality. Lemma 1. [3] Consider a set A that contains at least d + 1 points in general position in Rd. Let B and C be its disjoint subsets such that B = b, C = c, 2 b, c d and | | | | ≤ ≤ b + c d + 1. If the (b 1)-simplex formed by B and the (c 1)-simplex formed by ≥ − − C form a crossing pair, then the u-simplex (u b 1) formed by a point set B B ≥ − ′ ⊇ and the v-simplex (v c 1) formed by a point set C C satisfying B C = ∅, ≥ − ′ ⊇ ′ ∩ ′ B , C d and B , C A also form a crossing pair. | ′| | ′| ≤ ′ ′ ⊂ Proof. For the sake of contradiction, we assume that there exist a u-simplex and a v-simplex, formed respectively by the disjoint point sets B B and C C, ′ ⊇ ′ ⊇ that do not cross. We consider two cases. Case 1 Let us assume that Conv(B ) Conv(C ) = ∅.
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