Coloring \(d\) -Embeddable \(k\) -Uniform Hypergraphs

Discrete & Computational Geometry, Dec 2014

This paper extends the scenario of the Four Color Theorem in the following way. Let \(\fancyscript{H}_{d,k}\) be the set of all \(k\)-uniform hypergraphs that can be (linearly) embedded into \(\mathbb {R}^d\). We investigate lower and upper bounds on the maximum (weak) chromatic number of hypergraphs in \(\fancyscript{H}_{d,k}\). For example, we can prove that for \(d\ge 3\) there are hypergraphs in \(\fancyscript{H}_{2d-3,d}\) on \(n\) vertices whose chromatic number is \(\Omega (\log n/\log \log n)\), whereas the chromatic number for \(n\)-vertex hypergraphs in \(\fancyscript{H}_{d,d}\) is bounded by \({\mathcal {O}}(n^{(d-2)/(d-1)})\) for \(d\ge 3\).

A PDF file should load here. If you do not see its contents the file may be temporarily unavailable at the journal website or you do not have a PDF plug-in installed and enabled in your browser.

Alternatively, you can download the file locally and open with any standalone PDF reader:

https://link.springer.com/content/pdf/10.1007%2Fs00454-014-9641-2.pdf

Coloring \(d\) -Embeddable \(k\) -Uniform Hypergraphs

Carl Georg Heise 0 1 Konstantinos Panagiotou 0 1 Oleg Pikhurko 0 1 Anusch Taraz 0 1 0 K. Panagiotou Mathematisches Institut, Ludwig-Maximilians-Universitt Mnchen , Munich, Germany 1 O. Pikhurko Mathematics Institute and DIMAP, University of Warwick , Coventry, UK This paper extends the scenario of the Four Color Theorem in the following way. Let Hd,k be the set of all k-uniform hypergraphs that can be (linearly) embedded into Rd . We investigate lower and upper bounds on the maximum (weak) chromatic number of hypergraphs in Hd,k . For example, we can prove that for d 3 there are hypergraphs in H2d3,d on n vertices whose chromatic number is ( log n/ log log n), whereas the chromatic number for n-vertex hypergraphs in Hd,d is bounded by O(n(d2)/(d1)) for d 3. - The Four Color Theorem [1, 2] asserts that every graph that is embeddable in the plane has chromatic number at most four. This question has been one of the driving forces in Discrete Mathematics and its theme has inspired many variations. For example, the chromatic number of graphs that are embedabble into a surface of fixed genus has been intensively studied by Heawood [17], Ringel and Youngs [24], and many others. In this paper, we consider k-uniform hypergraphs that are embeddable into Rd in such a way that their edges do not intersect (see Definition 1 below). For k = d = 2 the problem specializes to graph planarity. For k = 2 and d 3 it is not a very interesting question because for any n N the vertices of the complete graph Kn can be embedded into R3 using the embedding (vi ) = i, i 2, i 3 i {1, . . . , n}. It is a well known property of the moment curve t (t, t 2, t 3) that any two edges between four distinct vertices do not intersect (see Proposition 14). As a consequence, we now focus our attention on hypergraphs, which are in general not embeddable into any specific dimension. Some properties of these hypergraphs (or more generally simplicial complexes) have been investigated (see e. g. [10,11,19,20, 27,31]), but to our surprise, we have not been able to find any previously established results which bound their chromatic number. However, Grnbaum and Sarkaria (see [15,26]) have considered a different generalization of graph colorings to simplicial complexes by coloring faces. They also bound this face-chromatic number subject to embeddability constraints. Before we can state our main results, we quickly recall and introduce some useful notation. We say that H = (V , E ) is a k-uniform hypergraph if the vertex set V is a finite set and the edge set E consists of k-element subsets of V , i. e. E Vk . For any hypergraph H , we denote by V (H ) the vertex set of H and by E (H ) its edge set. We define and call any hypergraph isomorphic to Kn(k) a complete k-uniform hypergraph of order n. Let H be a k-uniform hypergraph. A function : V (H ) {1, . . . , c} is said to be a weak c-coloring if for all e E (H ) the property |(e)| > 1 holds. The function is said to be a strong c-coloring if |(e)| = k for all e E (H ). The weak/strong chromatic number of H is defined as the minimum c N such that there exists a weak/strong coloring of H with c colors. The chromatic number of H is denoted by w(H ) and s(H ), respectively. Obviously, for graphs, weak and strong colorings are equivalent. We next define what we mean when we say that a hypergraph is embeddable into Rd . Here, aff denotes the affine hull of a set of points and conv the convex hull. Definition 1 (d-Embeddings) Let H be a k-uniform hypergraph and d N. A (linear) embedding of H into Rd is a function : V (H ) Rd , where ( A) for A V (H ) is to be interpreted pointwise, such that dim aff (e) = k 1 for all e E (H ) and conv (e1) conv (e2) = conv (e1 e2) for all e1, e2 E (H ). Table 1 Currently known lower bounds for the maximum weak chromatic number of a d-embeddable k-uniform hypergraph on n vertices as n The number in chevrons indicates the theorem number where we prove this bound The first property is needed to exclude functions mapping the vertices of one edge to affinely non-independent points. The second guarantees that the embedded edges only intersect in the convex hull of their common vertices. Note that the inclusion from left to right always holds. A k-uniform hypergraph H is said to be d-embeddable if there exists an embedding of H into Rd . Also, we denote by Hd,k the set of all d-embeddable k-uniform hypergraphs. One can easily see that our definition of 2-embeddability coincides with the classical concept of planarity [12]. Note that in general there are several other notions of embeddability. The most popular thereof are piecewise linear embeddings and general topological embeddings. A short and comprehensive introduction is given in Sect. 1 in [19]. Furthermore, there exist some quite different concepts of generalizing embeddability for hypergraphs in the literature, for example hypergraph imbeddings [32, Chap. 13]. We have decided to focus on linear embeddings, as they lead to a very accessible type of geometry and, at least in theory, the decision problem of whether a given k-uniform hypergraph is d-embeddable is decidable and in PSPACE [23]. One can show that the aforementioned three types of embeddings are equivalent only in the less than 3-dimensional case (see e. g. [3,4]), although piecewise linear and topological embeddability coincides if d k 2 or (d, k) = (3, 3), see [5]. Since piecewise linear and topological embeddings are more general than linear embeddings, all lower bounds for chromatic numbers can easily be transferred. Furthermore, we prove all our results on upper bounds for piecewise linear embeddings (and thus also for topological embeddings if d k 2 or (d, k) = (3, 3)) except for one case (namely Theorem 20). We can now give a summary of our main results in Tables 1 and 2, which contain upper or lower bounds for the maximum weak chromatic number of a d-embeddable k-uniform hypergraph on n vertices. All results which only follow non-trivially from prior knowledge are indexed with a theorem number from which they can be derived. Table 2 Currently known upper bounds for the maximum weak chromatic number of a d-embeddable k-uniform hypergraph on n vertices as n The number in chevrons indicates the theorem number where we prove this bound Considering the strong chromatic number, the question whether embeddability restricts the number of colors needed can be answered negatively by the following observation. Let n, d N such that d 3 and n d + 1 and let V = {1, . . . , n}. Let : R Rd , (x ) = (x , . . . , x d+1) be the (d + 1)-dimensional moment curve. Then (V ) are the vertices of a cyclic polytope P = conv (V ) (see [6,7,21]). As d 3, we have that P is 2-neighborly [13]. Define H ( P) = (V , E ( P)) to be the (d + 1)uniform hypergraph with E ( P) = {e V : e is the set of vertices of a facet of P}. Then H ( P) can be linearly embedded into Rd : for example, one can take the SchegelDiagram [28] of P with respect to some facet. Now, choose k N such that 2 k d + 1. Following [14, 7.1], for any hypergraph H = (W, E ), we call the k-shadow of H . As P is 2-neighborly we have that S2(H ( P)) = Kn and thus s(H ( P)) = n. Obviously, S2(Sk (H ( P))) = Kn and s(Sk (H ( P))) = n, too. Thus, we have demonstrated that for any 2 k d + 1 n there exists a k-uniform hypergraph on n vertices that is linearly d-embeddable and has strong chromatic number n. Thus, from now on, we restrict ourselves to the weak case and will always mean this when talking about a chromatic number. To conclude the introduction, here is a rough outline for the rest of the paper. In Sect. 2 the general concept of embedding hypergraphs into d-dimensional space is discussed. We also show the embeddability of certain structures needed later on, hereby extensively using known properties of the moment curve t (t, t 2, t 3, . . . , t d ). Then, Sect. 3 presents our current level of knowledge for the more difficult problem of weakly coloring hypergraphs. 2 Embeddability The first part of this section gives insight into the structure of neighborhoods of single vertices in a hypergraph H Hd,k . We will later use this information to prove upper bounds on the number of edges in our hypergraphs. This will then yield upper bounds on the weak chromatic number. However, we must first take a small technical detour into piecewise linear embeddings. As our hypergraphs are finite and of fixed uniformity we give a slightly simplified definition (for a more comprehensive introduction, see e. g. [25]). Definition 2 (Piecewise linear d-embeddings) Let H be a k-uniform hypergraph and D, d N. Let : V (H ) RD be a linear embedding of H and define (H ) = eE(H) conv (e). We say H is piecewise linearly embeddable if there exists : (H ) Rd such that is a homeomorphism from (H ) onto its image and there exists a (locally finite) subdivision K of (H ) (seen as a geometric simplicial complex) such that is affine on all elements of K . We call a piecewise linear embedding of H into Rd and we denote by HdP,kL the set of all piecewise linearly d-embeddable k-uniform hypergraphs. Note that such a always exists, as H H2k1,k by the MengerNbeling Theorem (see [20, p. 295] and [22]). Also, Definition 2 is independent of the choice of . Definition 3 (Neighborhoods) For a k-uniform hypergraph H and a vertex v V (H ) we say the neighborhood of v is NH (v)={w V (H ) : w = v and there is an edge in E (H ) incident with w and v}. We define the neighborhood hypergraph (or link) of v V (H ) to be the induced (k 1)-uniform hypergraph NHH (v) = (NH (v), {e\{v} : e E (H ), v e}) . Lemma 4 For a hypergraph H HdP,kL on n vertices, d k 2, and for any vertex v we have that NHH (v) HdPL1,k1. Proof Let d k 2, H HdP,kL, v V (H ), and Vv = NH (v) nonempty. Then there exist : V (H ) R2k1 a linear embedding and : (H ) Rd a piecewise linear embedding of H for some subdivision K of (H ) on whose elements is affine. Without restriction assume that (v) = 02k1 and (02k1) = 0d . Let Hv = (Vv {v}, {e E (H ) : v e}) be the sub-hypergraph of H of all edges containing v. Obviously, |(Hv) (the restriction of onto (Hv)) is a piecewise linear embedding of Hv for some subdivision Kv K . Let Kv1 = {e Kv : 02k1 e}. Then there exists an > 0 such that eKv1 i. e. all points in (Hv) are so close to 02k1 that they lie completely in elements of Kv that contain the origin. Then : Vv {v} R2k1, w (w) is a linear and thus | (Hv) a piecewise linear embedding of Hv for the subdivision Kv2 = {e (Hv) : e Kv1}. Let VKv2 (Vv) be the set of all subdivision points of Kv2 without 02k1 and let (x ) : x conv(e VK 2 ) for some e Kv2 . v (Hv). Then the image of ( |K ) lies completely in C \{x }. Finally, note that C \{x } is piecewise linearly homeomorphic to Rd1 [25, 3.20]. Let be such a (piecewise linear) homeomorphism . Then is a piecewise linear embedding of NHH (v) into Rd1 for some subdivision of K and NHH (v) HdPL1,k1. Note that it is quite plausible that a version of Lemma 4 for linear or general embeddings does not hold. Part (a) of the following result has previously been established by Dey and Pach for linear embeddings [8, Theorem 3.1]. Proof If k = 2, then (a) is equivalent to the fact that for G planar |E (G)| 3n 6. Given that (a) is true for some k 2, we show that (b) holds for k as well. Let H HkP+L1,k+1, v one of the n vertices. By Lemma 4, NHH (v) HkP,kL. By (a), (b) For a hypergraph H HkP+L1,k+1 on n vertices, k 2, and for any vertex v we have that |E (H )| 6nk1 12nk2 degH (v) 6nk1 12nk2 |E (NHH (v))| 6nk1 12nk2 degH (v) 6nk1 12nk2 |E (H )| = n(6nk1 12nk2) 6nk 12nk1 Corollary 6 For a hypergraph H HkP,kL on n vertices, k 3, and for any edge e E (H ) there exist at most k 6nk(2k112)!nk3 k other edges adjacent to it. Proof This follows from Lemma 5, since every edge has exactly k vertices and each of them has degree at most 6nk212nk3 . As e itself counts for the degree as well, one (k1)! can subtract k. We need to bound the number of edges in a d-embeddable hypergraph to prove upper bounds for the chromatic number. The following results will also help to do this. Note that there exist much stronger conjectured bounds (see [16, Conjecture 1.4.4] and [18, Conjecture 27]). Proposition 7 (Gundert [16, Proposition 3.3.5]) Let k 2. For a k-uniform hypergraph on n vertices that is topologically embedabble into R2k2, we have that |E (H )| < nk31k . 2, we have that (k Proof This follows from inductively applying Lemma 4 and Proposition 7. edge e E (H ) there exist at most (k(k+1)2!)! nk13 1k k other edges adjacent to it. 3, and for any Proof This fact follows analogously to Corollary 6 from Corollary 8. Theorem 10 (Dey and Pach [8, Theorem 2.1]) Let k 2. For a k-uniform hypergraph on n vertices that is linearly embedabble into Rk1, we have that |E (H )| < kn (k1)/2 . Corollary 11 For a hypergraph H Hk1,k on n vertices, k 2, and for any edge e E (H ) there exist at most kn (k1)/2 1 other edges adjacent to it. Proof This fact follows obviously from Theorem 10. In order to find lower bounds for the chromatic number of hypergraphs later on, we need to be able to prove embeddability. The following theorem from Shephard will turn out to be very useful when embedding vertices of a hypergraph on the moment curve. Theorem 12 (Shephard [29]) Let W = {w1, . . . , wm } Rd be distinct points on the moment curve in that order and P = conv W . We call a q-element subset {wi1 , wi2 , . . . , wiq } W with i1 < i2 < < iq contiguous if iq i1 = q 1. Then U W is the set of vertices of a (k 1)-face of P if and only if |U | = k and for some t 0 U = YS X1 Xt YE , where all Xi , YS , and YE are contiguous sets, YS = or w1 YS, YE = or wm YE , and at most d k sets Xi have odd cardinality. Shephards Theorem thus says that the absolute position of points on the moment curve is irrelevant and only their relative order is important. Furthermore, note that all points in W are vertices of P. The following corollary helps in proving that two given edges of a hypergraph intersect properly. Corollary 13 In the setting of Theorem 12 assume that W = U1 U2 where U1 and U2 are embedded edges of a k-uniform hypergraph. Then these edges do not intersect in a way forbidden by Definition 1, if there exists j {1, 2} such that U j = YS X1 Xt YE Proof The two edges U1 and U2 do not intersect in a way forbidden by Definition 1 if at least one of them is a face of P = conv W , which is the case for U j . Proposition 14 Let A, B, C , and D be four distinct points on the moment curve in R3 in arbitrary order. Then the line segments A B and C D do not intersect. In the k = d = 3 case Corollary 13 allows zero odd sets Xi . Thus, we can easily classify all possible configurations for two edges. Lemma 15 Let H be a 3-uniform hypergraph and : V (H ) R3 such that maps all vertices one-to-one on the moment curve and for each pair of edges e and f sharing at most one vertex, the order of the points (e f ) on the moment curve has one of the Configurations 112 shown in Table 3. Then is an embedding of H . Table 3 Possible configurations for two edges e and f on the moment curve in R3 sharing at most one vertex The vertices of e\ f are marked with E, those of f \e marked with F, and a joint vertex is marked with I. Equivalent cases, one being the reverse of the other, are only displayed once Proof Note that the relative order of edges with two common vertices is irrelevant as they always intersect according to Definition 1. Configurations 111 follow directly from Corollary 13 for k = d = 3. Thus, we are left with Configuration 12 and it is sufficient to prove the following: For x0,0 < x1,0 < x0,1 < x2,0 < x1,1 < x2,1 R, : R R3, (x ) = (x , x 2, x 3) the moment curve, and Di = {x0,i , x1,i , x2,i } we have that conv (D0) conv (D1) = . Assume otherwise. Note that if two triangles intersect in R3 the intersection points must contain at least one point of the border of at least one of the triangles. Thus, without loss of generality, conv{ (x j1,0), (x j2,0)} conv (D1) = . However, by Theorem 12 we know that conv{ (x j1,0), (x j2,0)} is a face of the polytope P = conv({ (x j1,0), (x j2,0)} (D1)) which is a contradiction. Note that if we have two edges with vertices on the moment curve as in Configurations 1316 they generally do intersect in a way forbidden by Definition 1. Also, we have presented above all possible cases for the relative order of vertices of two edges on the moment curve. Not all of them will actually be needed in the proofs of the next section. 3 Bounding the Weak Chromatic Number For d, k, n N we define dw,k (n) = max{ w(H ) : H Hd,k , |V (H )| = n} to be the maximum weak chromatic number of a d-embeddable k-uniform hypergraph on n vertices. In this section, we give lower and upper bounds on dw,k (n). Obviously, dw,k (n) is monotonically increasing in n and in d and monotonically decreasing in k if the other parameters remain fixed. Proof Let H H2,3 and V = V (H ). Then G = S (H ) is a planar graph, thus (G) 4. Let : V {1, 2, 3, 4} be a 4-coloring of G. Define In any triangle {u, v, w} of H under the coloring these vertices have exactly three different colors. Therefore, under the coloring at least one vertex with color 1 and one vertex with color 2 exists. Thus is a valid 2-coloring of H . Theorem 18 Let d 3. Then one has (d 1)! = O n dd21 as n . This result also holds for piecewise linear embeddings. Proof Let H HdP,dL Hd,d . By Corollary 6 we know that every edge is adjacent to at most = d(6nd2 12nd3)/(d 1)! d other edges. We want to apply the Lovsz Local Lemma [9,30] to bound the weak chromatic number of H . Let c N. In any c-coloring of the vertices of H an edge is called bad if it is monochromatic and good if not. In a uniformly random c-coloring the probability 1 . Moreover, let e be any edge in H and F be for any one edge to be bad is p = cd1 the set of edges in H not adjacent to e. Then the events of e being bad and of any edges from F being bad are independent. Thus the event whether any edge is bad is independent from all but at most other such events. The Lovsz Local Lemma guarantees us that with positive probability all edges are good if e p ( + 1) 1. This implies that H is weakly c-colorable. Note that + 1) 1 ed(6nd2 12nd3) (d 1)! ed + e cd1. Choosing an integer c (d 1)! ed(6nd2 12nd3) (d 1)! ed + e the hypergraph H is c-colorable and w(H ) c. Theorem 19 Let d 3. Then one has This result also holds for piecewise linear embeddings. = O n1 3 d11d as n . Theorem 20 Let d 2. Then one has Proof By Corollary 9 we know that every edge is adjacent to at most = (d(d+1)2!)! nd13 1d d other edges. The rest of the proof is now analogous to the proof of O n1/2+1/(2d2) if d is even as n . Proof By Corollary 11 we know that every edge is adjacent to at most = dn (d1)/2 1 other edges. The rest of the proof is now analogous to the proof of Theorem 18. By monotonicity, the upper bounds presented here also hold if the uniformity of the hypergraph is larger than stated in Theorems 18 and 19. In the remaining part of this section, we now consider lower bounds for the weak chromatic number of hypergraphs. Theorem 21 For n 2 we have as n . Proof We first define a sequence of hypergraphs Hm for m 2 such that w(Hm ) m. Set H2 = K (3) which has 3 vertices. Define Hm for m > 2 iteratively, assuming 3 w(Hm1) m 1. Take m new vertices {v0, . . . , vm1} and m(m 1)/2 disjoint copies of Hm1, labeled H m[0,11], . . . , H [m2,m1]. m1 The edges of Hm shall be all former edges of all H m[i,j1] together with all edges of Relative order of v Additional condition Case number 1 6 12 7 2 8 3 1 Relative order of i1, i2, j1, j2 Fig. 1 Construction of Hm Table 4 Sub-cases of Case 3 in the proof of Theorem 21 referring to the corresponding cases of Lemma 15 Table 5 Sub-cases of Case 4 in the proof of Theorem 21 referring to the corresponding cases of Lemma 15 Thus, the order given by fm provides an embedding of Hm . To estimate nm , we use the following recursion n2 = 3, nm = m + nm1 m(m 1)/2 for m > 2. This can be bounded by nm m2m =: nm . Then and we finally get that Note that by monotonicity also log nm log m log log n m = 2m log(2m log m) 2m log nm log nm m 2 log log nm 2 log log nm Theorem 22 Let d 3. For n d we have as n . by vertices on the moment curve with edges intersecting according to Corollary 13 (or Lemma 15 if d = 4). 2 = K (d). The hypergraph H2d has d vertices, one edge, and is weakly Let H d d 2-colorable. Define Hmd for m > 2 iteratively, given that w(Hmd1) m 1. For that, take one copy of H d m1 and one copy of (d 1)-uniform Hmd1. The edges of Hmd shall be all edges of H d m1 and all edges of the form ({v} e) for v V (Hmd1) and e E (Hmd1). Assume that there exists a weak (m 1)-coloring of Hmd . Then there has to be at least one monochromatic edge e E (Hmd1). No vertex of H d m1 can be colored with this color, so its edges must be weakly (m 2)-colored. This is a contradiction and thus w(Hmd ) m. We now claim that Hmd H2d3,d for all m 2. As in the proof of Theorem 21, we give a function fm(d) : V (Hmd ) {1, . . . , n(md)} where n(md) is the number of vertices of Hmd . This defines the order in which the vertices of Hmd will be arranged on the moment curve t (t, . . . , t 2d3). We then use Corollary 13 to prove that Hmd is embeddable Fig. 2 Construction of Hmd via arbitrary points on the moment curve. As before, the absolute position of vertices on the moment curve is not important. For a fixed uniformity d and dimension 2d 3, Corollary 13 guarantees that if for two given edges the vertices of at least one edge have at most d 3 odd contiguous subsets, they intersect properly according to Definition 1. For d = 3 we can set fm(3) = fm for all m 2, where fm is as in the proof of Theorem 21. For d > 3 we have by assumption that there exists a corresponding family of functions fm(d1) : V (Hmd1) {1, . . . , n(md1)} m such that the vertices of Hmd1 arranged in that order on the moment curve form an embedding. We then have to give an appropriate family of functions fm(d) for d. H2d can be embedded into R2d3 via any d points on the moment curve, so f2(d) : V (H2d ) {1, . . . , d} can be chosen arbitrarily. Assume that fm(d)1 has already been defined and gives an embedding of Hmd1. We define fm(d)(v) = fm(d)1(v) for v V (H d m1 + fm(d1)(w). This is m1) and for any w V (Hmd1) we set fm(d)(w) = n(d) also shown in Fig. 2. Arrange the vertices of Hmd on the moment curve in that order and pick any two edges g1 and g2. Case 1: Both edges are from the subhypergraph H d m1. Then they intersect in accordance to Definition 1 and Corollary 13 as their relative order reflects that of f (d) . m1 Case 2: One edge is from H d m1 and the other of the form ({v} e) where v V (Hmd1) and e E (Hmd1). Then both edges have at most one odd contiguous subset (besides the first and last one), which is no problem for d > 3. Case 3: g1 = ({v1} e1) and g2 = ({v2} e2). Then the edges e1 and e2 intersect according to Corollary 13 (or Lemma 15 if d = 4) and g1 and g2 have at most one more odd contiguous subset than the edges e1 and e2 had in the ordering of fm(d1). The last number, by assumption, was bounded from above by (d 1) 3 for at least one ei , i {1, 2} (unless d = 4 and they intersect according to Case 12 in Table 3, see below). So at least one gi has at most d 3 odd contiguous subsets. Thus, the order given by fm(d) provides an embedding of Hmd . Note that there is one small exception to Case 3 when d = 4. Here, e1 and e2 could be in the relative position of Case 12 in Table 3 and consequently have more than (d 1) 3 = 0 odd contiguous subsets. However, this is no problem as in all possible extensions to g1 and g2 at least one of the edges continues to have only one odd contiguous subset (see Table 6). To bound the number of vertices of Hmd we use Iteratively, we get that n(md) = d + m2m+d3 and thus rm=3 nr(d1) m n(md1) md3 n m = log(m) log log n(md) (2m + d 3) log ((2m + d 3) log(m)) 2m + d 3. I E E F E F F Note that by monotonicity also m 2 log log n(md) 4 Conclusions and Open Questions Starting from the Four Color Theorem we have shown that it has no direct analogon for higher dimensions in general. Rather, in almost all cases, the number of colors needed to color a hypergraph embedabble in a certain dimension is unbounded. However, some questions still need to be answered. Firstly, it would be very interesting to see whether the logarithmic-polynomial difference between lower and upper bounds for the weak coloring case can be improved substantially. If the conjectures by Gundert and Kalai mentioned in Sect. 2 were true, the upper bound for weak colorings could be lowered as follows. Conjecture 23 Let k 1 d 2k 2. Then one has as n . Further, in the weak coloring case, for k = d + 1 no examples with an unbounded number of colors needed have yet been found and a finite bound is still possible. Also, the question whether the maximum chromatic number for some fixed k, d , and n actually differs for linear and piecewise linear embeddings, remains an open problem. Acknowledgments The authors wish to thank Penny Haxell for helpful discussions. They also would like to thank two anonymous referees for careful and valuable remarks concerning the presentation of this work, in particular considerably simplifying the treatment of strong colorings. Carl Georg Heise was partially supported by the ENB graduate program TopMath and DFG Grant GR 993/10-1. The author gratefully acknowledges the support of the TUM Graduate Schools Thematic Graduate Center TopMath at the Technische Universitt Mnchen. Oleg Pikhurko was partially supported by the Engineering and Physical Sciences Research Council (grant EP/K012045/1), the Alexander von Humboldt Foundation, and the European Research Council (Grant No. 306493). Anusch Taraz was partially supported by DFG Grant TA 309/2-2. Open Access This article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution, and reproduction in any medium, provided the original author(s) and the source are credited.


This is a preview of a remote PDF: https://link.springer.com/content/pdf/10.1007%2Fs00454-014-9641-2.pdf

Coloring \(d\) -Embeddable \(k\) -Uniform Hypergraphs, Discrete & Computational Geometry, 2014, 663-679, DOI: 10.1007/s00454-014-9641-2