Tverberg-Type Theorems for Separoids

Discrete & Computational Geometry, Feb 2006

Juan Jose Montellano-Ballesteros, Attila Por, Ricardo Strausz

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Tverberg-Type Theorems for Separoids

Discrete Comput Geom Geometry Discrete & Computational Juan Jose´ Montellano-Ballesteros 1 Attila P o´r 0 Ricardo Strausz 0 1 0 Institut Teoreticke ́ Informatiky, Universita Karlova v Praze , Malostranske ́ na ́m. 25, Praha 1 , Czech Republic 1 Instituto de Matema ́ticas, U.N.A.M., Circuito Exterior, C.U. , 04510 Me ́xico D.F., Me ́xico Let S be a d-dimensional separoid of (k − 1)(d + 1) + 1 convex sets in some “large-dimensional” Euclidean space IEN . We prove a theorem that can be interpreted as follows: if the separoid S can be mapped with a monomorphism to a d-dimensional separoid of points P in general position, then there exists a k-colouring ς : S → Kk such that, for each pair of colours i, j ∈ Kk , the convex hulls of their preimages do intersect-they are not separated. Here, by a monomorphism we mean an injective function such that the preimage of separated sets are separated. In a sense, this result is “dual” to the Hadwiger-type theorems proved by Goodman and Pollack (1988) and Arocha et al. (2002). We also introduce ϑ (k, d), the minimum number n such that all d-dimensional separoids of order at least n can be k-coloured as before. By means of examples and explicit colourings, we show that for all k > 2 and d > 0, Introduction and Statement of Results - (d + 1) + 1. k 2 Furthermore, by means of a probabilistic argument, we show that for each d there exists a constant C = C (d) such that for all k, ϑ (k, d) ≤ C k log k. As suggested by Danzer et al. (1963) [4], the relationship between Helly’s, Radon’s and Carathe´odory’s theorems “could be best understood by formulating various axiomatic settings for the theory of convexity”. The first attempt to give such an axiomatic setting was made by Levi (1951) [ 11 ], who uses Helly’s theorem (1923) [ 10 ] as a starting point. More recently, the concept of a separoid was introduced [1], [3], [12], [13], [17]–[19] as a new attempt in this direction that is instead based on Radon’s theorem (1921) [ 15 ]. A separoid is a (finite) set S endowed with a symmetric relation † ⊂ 22S defined on its family of subsets, which satisfies the following properties for all A, B ⊆ S: (i) A † B ⇒ A ∩ B = ∅, (ii) A † B and B ⊂ B (⊆ S\ A) ⇒ A † B . A pair A † B is called a Radon partition. Each part ( A and B) is called a component and the union A ∪ B is the support of the partition. The (combinatorial) dimension of S, denoted by d(S), is the minimum d such that every subset of S with at least d + 2 elements is the support of a Radon partition. By the second condition, the minimal Radon partitions determine the separoid. A pair of disjoint subsets α, β ⊆ S that is not a Radon partition is said to be separated, and denoted by α | β (see [ 1 ]). Now, given a family of convex sets F = {C1, . . . , Cn} in some Euclidean space IEd , a separoid S(F ) on {1, . . . , n} can be defined by the following relation: for all α, β ⊆ S(F ), α | β ⇐⇒ i∈α j∈β Ci ∩ Cj = ∅, where · denotes the convex hull. Analogously, the Radon partitions are defined by A † B ⇐⇒ A ∩ B = ∅ and i∈A j∈B Ci ∩ Cj = ∅. Conversely, as proved by Arocha et al. [1], every (abstract) separoid can be represented in such a way by a family of convex sets in some Euclidean space. Therefore each separoid S has a minimum dimension where it can be represented called the geometric dimension of S, denoted by gd(S). Furthermore, as proved by Strausz [ 19 ], if the separoid S is acyclic (i.e., if ∅ | S), then gd(S) ≤ |S| − 1 (see also [ 17 ]). The following theorem is an easy corollary of Tverberg’s theorem [20] (see also [5] and the references therein). Theorem 1. Let S be a separoid of order |S| = (k − 1)(d + 1) + 1, where d = gd(S). Then there exists a k-colouring ς : S → Kk = {1, . . . , k} such that every pair of chromatic classes is not separated; i.e., the preimage of every pair of colours i, j ∈ Kk , is a Radon partition ς −1(i ) † ς −1( j ). Indeed, a stronger conclusion can be reached. Represent the separoid S with convex sets in IEd , where d = gd(S). If we choose a point in each convex set and apply Tverberg’s theorem to this set of points, then we can find a k-colouring of S such that there is a point that is in the convex hull of every chromatic class. Following [18], if the conclusion of Theorem 1 holds, we say that there exists a chromomorphism onto the complete separoid Kk of order k (to be defined; see Fig. 1). If such a chromomorphism exists for a given S, we write S −→ Kk ; otherwise we write S −→ Kk . In this note we are interested in purely combinatorial conditions that guarantee the existence of chromomorphisms onto complete separoids. As shown by Fig. 2, in Theorem 1 the geometric dimension cannot be replaced by the combinatorial dimension without adding a new ingredient—observe that d(S) ≤ gd(S). Thus, while replacing gd(S) by d(S), we may add a Hadwiger-type hypothesis that allows us to prove the following: Theorem 2. Let S be a d-dimensional separoid of order |S| = (k − 1)(d + 1) + 1. Suppose that in addition, there exists a d-dimensional separoid of points P in general position and a monomorphism from S into P (i.e., an injective function µ : S → P such that the preimage of separated sets are separated). Then S −→ Kk . Arocha et al. [1] proved a Hadwiger-type theorem that, supposing the existence of a monomorphism “from the left” ν: P → S, concludes the existence of a virtual transversal. That is, there are “as many” hyperplanes transversal to the family as there are hyperplanes through an -flat (e.g., while the family in Fig. 1(b) has a 0-transversal, that of Fig. 1(a) has a virtual 0-transversal). This result extends ideas from Goodman and Pollack [7] who used the notion of order type to characterise the existence of hyperplane transversals. On the other hand, Theorem 2 supposes the existence of a monomorphism “to the right”, and concludes that there is a virtual Tverberg partition (i.e., a partition with a virtual 0-transversal). Thus these theorems may be seen as “dual”—at least in the case = d(S). However, the Hadwiger-type hypotheses are “geometric” in nature; that is, they restrict the convex sets that represent the separoid to be in some “special position”. (See [ 4 ] for the early work on such “special position” hypotheses, and see [ 5 ] and [ 8 ] for excellent updates on the subject.) The following questions arise. How far can the hypothesis of Theorem 2 be weakened without changing the conclusion? Is there a purely combinatorial Tverberg-type theorem? We now introduce the following new concept. The (k, d)-Tverberg number ϑ (k, d) is the minimum number n ∈ IN such that every d-dimensional separoid of order at least n maps onto Kk with a chromomorphism; that is, ϑ (k, d) is minimal with the property |S| ≥ ϑ (k, d(S)) ⇒ Analogously, if S denotes a class of separoids, we denote by ϑS (k, d) the (k, d)-Tverberg number restricted to the class S. Thus, Tverberg’s theorem can be rewriten as ϑP (k, d) = (k − 1)(d + 1) + 1, where P denotes the class of separoids of points. Analogously, using the notion of pseudoconfiguration of points, Roudneff [ 16 ] proved that ϑM(k, 2) = 3k − 2, where M denotes the class of oriented matroids. In this direction we prove the following: Proposition 1. If G denotes the class of (simple) graphs—thought of as separoids whose minimal Radon partitions are pairs of singletons—then ϑG (k, d) ≤ (k − 1)(d + 1) + 1. (Observe the close relation between ϑG (k, d) and the so-called pseudoachromatic number [ 9 ].) However, in general the (k, d)-Tverberg number is greater than that. Indeed, we will prove that Theorem 3. For all pairs of natural numbers k > 2 and d > 0 it follows that (k − 1)(d + 1) + 1 < ϑ (k, d) < k 2 (d + 1) + 1. Furthermore, by means of a probabilistic argument, we will prove that Theorem 4. For each d > 0, the constant C = 2d+4 is such that for all k ≥ d + 2, ϑ (k, d) ≤ C k log k. 2. Definitions and Proof of Theorem 2 In order to be self-contained, we start with some basic notions and examples. Every (finite and acyclic) separoid S can be represented by a family of (convex) polytopes in the (|S| − 1)-dimensional Euclidian space [ 17 ], [ 19 ]. The construction is as follows. Let S be identified with the set {1, . . . , n}. For each element i ∈ S and each minimal Radon partition A † B such that i ∈ A, consider the point i 1 ρA†B = ei + 2 1 |B| b∈B eb − 1 | A| a∈A ea , where ei denotes the i th vector of the canonical basis of IRn. Then each element i is represented by the convex hull of all such elements: i → Ki = ρiA†B : i ∈ A and A † B . Observe that the convex sets Ki live in the affine hyperplane spanned by the basis. It is simple to verify that this construction is correct and that the implicit bound n − 1 is tight. Thus there is a minimum dimension in which S can be represented, called the geo metric dimension of S and denoted by gd(S). Furthermore, if the separoid can be represented by a family of points in some Euclidian space, it is called a point separoid [ 3 ], [ 12 ] (also known as a linear oriented matroid [ 2 ] or as an order type [ 6 ]). The order of the separoid S is the cardinal |S| and its size is the cardinal | † | (i.e., the number of Radon partitions). The separoid of order d + 1 and size 0 is called the d-dimensional simploid; that is, a separoid is a simploid if every subset is separated from its complement. Simploids can be represented by the vertex sets of simplices—hence the name. The (combinatorial) dimension of a separoid S is the maximum dimension of its induced simploids and is denoted by d(S) (observe that this definition is equivalent to that given in the Introduction). We say that the separoid is in general position if every set of d(S) + 1 elements induces a simploid. Thus, a d-dimensional separoid of convex sets is in general position if (and only if) every d + 2 elements admit a d-flat transversal but no d + 1 elements do. Furthermore, d is the minimum number with that property. A separoid is called a Radon separoid if each minimal Radon partition is unique in its support; i.e., if A † B, C † D are minimal then A ∪ B ⊆ C ∪ D ⇒ { A, B} = {C, D}. Observe that if S is a point separoid, then d(S) = gd(S) and it is a Radon separoid. Furthermore, a separoid S in general position is a point separoid if and only if d(S) = gd(S) (see [ 3 ]). The (acyclic) separoid K is complete if for all i, j ∈ K we have that i † j ; i.e., if its size is as big as possible. We denote by Kk the complete separoid of order k. Observe that a separoid is complete if and only if its dimension is zero. Given two separoids S and P, a function ϕ: S → P is a morphism if the pre image of separations are separations (see [ 1 ] and [ 17 ] for several important examples of morphisms); that is, for all α, β ⊆ P, α | β ⇒ ϕ−1(α) | ϕ−1(β). If the function ϕ is injective (resp. surjective), the morphism is called a monomorphism (resp. an epimorphism). An epimorphism is a chromomorphism if the preimage of minimal Radon partitions are Radon partitions. The main example to have in mind while thinking about chromomorphisms is the following—it motivates the name of such morphisms. Consider a family of convex sets S = {C1, . . . , Cn}. Given an (effective) k-colouring ς : S → {1, . . . , k}, let Di = ς −1(i ) be the convex hull of the union of those convex sets coloured i , for i = 1, . . . , k. Let T = {D1, . . . , Dk }. The induced function, also denoted by ς : S −→ T , is a chromomorphism between those separoids. Given a (simple and undirected) graph G = (V , E ), a separoid S on V can be defined with the relation, for i, j ∈ V , i † j is minimal ⇐⇒ i j ∈ E . Indeed, this definition induces a functoral embedding from the category of graphs into that of separoids when both classes are endowed with homomorphisms (see [ 13 ]). Conversely, given a separoid S, we say that S is a graph if, for A, B ⊆ S, A † B is minimal ⇒ | A||B| = 1. Clearly Kk is the complete graph of order k—hence the notation. Observe that a graph H is a minor of a connected graph G if and only if there exists a chromomorphism G −→ H with all its fibres connected. Proof of Proposition 1. Let G be a d-dimensional graph. We need to prove that |G| ≥ (k − 1)(d + 1) + 1 ⇒ G −→ Kk . Denote by α(G) = d + 1 the independence number and by χ (G) the chromatic number. Using the well-known Erdo˝s inequality, |G| ≤ χ (G)α(G), we have that (k − 1)α(G) + 1 ≤ |G| < χ (G)α(G) + 1, which implies that k ≤ χ (G). Observe that any homomorphism—or proper colouring if you will— ϕ: G → Kχ(G) is also a chromomorphism. Furthermore, for all n ≤ m there is a chromomorphism Km −→ Kn. Therefore, there is a chromomorphism ψ : Kχ(G) −→ Kk and we have that ς = ψ ◦ ϕ is the desired chromomorphism. Proof of Theorem 2. Let S be a d-dimensional separoid of order (k − 1)(d + 1) + 1. Suppose there is a monomorphism µ : S → P into a d-dimensional point separoid in general position. Due to Tverberg’s theorem, there exists a chromomorphism τ : P −→ Kk . We now show that ς = τ ◦ µ is a chromomorphism. Let i † j be an edge of Kk . Since τ is a chromomorphism, we have that τ −1(i )†τ −1( j ). Then there exist A ⊆ τ −1(i ) and B ⊆ τ −1( j ) such that A † B is minimal. Since P is in general position, | A ∪ B| = d + 2. Since µ is injective, |µ −1( A ∪ B)| = d + 2 and there exist C † D such that C ∪ D = µ −1( A ∪ B). Therefore, since µ is a monomorphism, µ( C ) † µ( D). Since P is a point separoid, it is a Radon separoid and we may suppose that µ( C ) = A and µ( D) = B. Finally, since C ⊆ ς −1(i ) and D ⊆ ς −1( j ), we have that ς −1(i ) † ς −1( j ), which concludes the proof. Figure 3 shows that the hypothesis of general position cannot be dropped without adding a new ingredient. On the other hand, if we suppose—as did Goodman and Pollack [ 7 ] and Arocha et al. [ 1 ]—that the monomorphism comes “from the left” µ : P → S, then such a hypothesis is not needed and the argument is much simpler (see the proof of Lemma 1). Observe that Fig. 3 also shows that the existence of a virtual line does not imply the existence of the corresponding chromomorphism. 3. Proofs of Theorems 3 and 4 We start this section with a simple, but useful, structural result that allows us to restrict our attention to Radon separoids in general position. Lemma 1. Given a d-dimensional separoid S, there exists a d-dimensional Radon separoid R in general position such that R −→ Kk ⇒ Proof. Let R be defined on the same set as S, and with the following set of minimal Radon partitions: for each subset X ∈ d +S2 , choose a single Radon partition A † (X \ A) of S to be in R. Clearly R is a Radon separoid in general position and d(R) = d(S). Furthermore, the identity map ν: R → S is a monomorphism. Now, suppose that ς : R −→ Kk is a chromomorphism; that is, suppose that for each edge i † j of Kk it follows that ς −1(i ) † ς −1( j ). Since ν is a monomorphism, we have that ν ◦ ς −1(i ) † ν ◦ ς −1( j ). Therefore, ς ◦ ν−1: S −→ Kk is a chromomorphism which concludes the proof. In what follows, given a partition X1, . . . , Xk we say that it has type (|X1|, . . . , |Xk |). In particular, given a Radon partition A † B we denote its type as the pair (| A|, |B|). Theorem 3 follows immediately from the following two lemmas. Lemma 2. For all k > 2 and d > 0 there exists a d-dimensional separoid S of order |S| = (k − 1)(d + 1) + 1 such that S −→ Kk . Proof. Let S = {0, 1, . . . , (k − 1)(d + 1)} be endowed with the following minimal S Radon partitions: for each A ∈ d+2 let x † ( A\x ), where x ∈ A is chosen such that x = 0  1 a = 0 a if 0, 1 ∈ A, if 0 ∈ A and if 0 ∈ A and if 0, 1 ∈ A. 1 ∈ A, 1 ∈ A, Since each minimal Radon partition has type (1, d + 1), if S −→ Kk then the induced partition must have type (1, d + 1, . . . , d + 1). That isolated element in the partition is called the singleton. Now, suppose that S −→ Kk and look at the partition induced by such a colouring. The singleton cannot be 0 because 0 †( A\0) only if 1 ∈ A, and 1 can only be in one part. The singleton cannot be 1 because 1 †( A\1) only if 0 ∈ A and 0 must be in some part. Thus suppose a ∈ {0, 1} is the singleton. However, then a | [[1]], where [[1]] denotes the chromatic class of 1; this is a contradiction. Lemma 3. Let S be a d-dimensional separoid. Then S −→ Kk . Proof. Let H = Hk denote the graph resulting from deleting an edge to Kk ; that is, H is the set (of colours) {1, . . . , k} and for each pair i j ∈ k2 except for one, say 1k, there is an edge i † j . Let S = Si j be a partition of type (d +2, d +1, . . . , d +1, d). Furthermore, suppose that |S12| = d +2 and |S1k | = d. Below we exhibit a chromomorphism ς : S\S1k −→ H . Then we extend it to a chromomorphism onto Kk using the remaining d elements of S. Since |S12| = d + 2, there is a Radon partition A † B whose support is S12. Assign colours respectively (i.e., let ς ( A) = 1 and ς (B) = 2). We may suppose that A has the maximum size that a component may have Remark. in S\S1k . Choose any element of colour 1, say a ∈ A. We now use a to extend the colouring to the parts S1 j (with 1 < j = k) so that ς becomes onto the edges incident to 1. That k is, for each pair 1 j ∈ 2 \1k, the set S1 j ∪ a, which consists of d + 2 elements, defines a Radon partition A † B (we may suppose a ∈ A ). Thus, we can assign colours by ς ( A ) = 1 and ς (B ) = j . Now, choose an element coloured 2, say b ∈ ς −1(2), and use it to extend all parts S2 j , with 2 < j (i.e., consider S2 j ∪ b, take its Radon partition and assign colours). Then repeat for colour 3, colour 4 and so on. At the end of such a process the colouring ς is the desired chromomorphism. Trying to extend ς to a chromomorphism onto Kk , we meet one possible obstruction: if we choose two elements coloured 1, say a, b ∈ ς −1(1), and consider the set S1k ∪{a, b} of d + 2 elements, then the defined Radon partition A † B is such that a ∈ A and b ∈ B (analogously if both elements are coloured k). Also, if we choose one element of each colour, say a ∈ ς −1(1) and b ∈ ς −1(k), and consider S1k ∪ {a, b}, then the respective Radon partition A † B contains both elements on the same component, say {a, b} ⊂ A . However, by the Remark, we may suppose that |ς −1(1) ∪ ς −1(k)| ≥ d + 2. Then there is a minimal Radon partition C † D whose support is contained in the preimage of this “missing edge”. Furthermore, since ς −1(1) | ς −1(k), there exist a pair of elements a ∈ ς −1(1) and b ∈ ς −1(k) which appears on the same side of that Radon partition, say {a, b} ⊂ C . Therefore, we can apply the previous method, but starting with S12 = C ∪ D, to find another chromomorphism ς : S\S1k −→ H such that ς (C ) = 1. Then ς can be extended to a chromomorphism S −→ Kk . Given a separoid S of order 2t k, we denote by the set of all k-colourings of S such that each chromatic class consists of exactly 2t elements. Analogously, given a Radon separoid T of order 2t+1 in general position we denote by the set of all 2-partitions of T into two sets of order 2t . Furthermore, a pair (α, β) ∈ is called a halving of T . We denote by pT the probability that α | β, for a randomly and uniformly choosen (α, β) ∈ . The proof of Theorem 4 is mainly based on the following: Lemma 4. If T is a d-dimensional Radon separoid of order 2t+1 in general position, then pT ≤ (d + 2)/22t−d−1 . Before proving this lemma, we see how it is used. Proof of Theorem 4. Let t = d + 4 + log log k and let S be a d-dimensional Radon separoid of order 2t k. Since 2t ≤ 2d+4 log k it is enough to prove that S −→ Kk . Let S = S1 ∪ · · · ∪ Sk be a random partition of S. Let Ei, j ⊂ be the event that Si | Sj . We claim that \ It is clear that all events Ei, j have the same probability. Now, we can obtain S1 and S2 as follows. Randomly choose a set T ⊂ S of order 2t+1 and let (S1, S2) ∈ be a random halving of T. By Lemma 4, the probability that S1 | S2 is at most (d + 2)/22t−d−1 . Therefore  Prob   Ei, j  ≤ (d + 2)k−4 < 1. k2 (d + 2)2−22+loglogk ≤ 2 Proof of Lemma 4.. We prove the lemma by induction on d. If d = −1 and t ≥ 0 then eavnedrysueplpeomseentthoefleTmims anoist tsreupeafroarteadllfdrom<tdhe.Iefmt p≤tydsewt.eThhavues tphTat=(d0+.N2o)/w2,2lt−edt−d1 >≥10; pT = Prob[E ] ≤ Prob[E1] + Prob[E \E1] ≤ Prob[E1] + Prob [E \E1] Prob E 1 ≤ (d + 2)/22t−d−1 . Therefore, Observe the relation with Rado’s original bound on the “Tverberg’s number” [14]: ϑP ≤ (k − 2)2d + d + 2. 1. J.L. Arocha , J. Bracho , L. Montejano , D. Olivero and R. Strausz . Separoids: their categories and a Hadwiger-type theorem for transversals . Discrete Comput. Geom. , 27 ( 2002 ), 377 - 385 . 2. A. Bjo¨rner, M. Las Vergnas , B. Sturmfels , N. White and G.M. Ziegler . Oriented Matroids. Encyclopedia of Mathematics and its Applications , vol. 43 . Cambridge University Press, Cambridge, 1993 . 3. J. Bracho and R. Strausz . 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Juan Jose Montellano-Ballesteros, Attila Por, Ricardo Strausz. Tverberg-Type Theorems for Separoids, Discrete & Computational Geometry, 2006, 513-523, DOI: 10.1007/s00454-005-1229-4