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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 R d 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 K n can be embedded into R 3 using the embedding 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, Grünbaum 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 ⊆ V k . 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 K (k) n a complete k-uniform hypergraph of order n.
Let H be a k-uniform hypergraph. A function κ : V (H ) → {1, . . . , c} is said to be a weak c-coloring if for all e ∈ E(H ) the property |κ(e)| > 1 holds. The function κ is said to be a strong c-coloring if |κ(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 χ w (H ) and χ 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 R d . Here, aff denotes the affine hull of a set of points and conv the convex hull.  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 R d . Also, we denote by H d,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. 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 Then ϕ(V ) are the vertices of a cyclic polytope P = conv ϕ(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 R d : for example, one can take the Schegel-Diagram [28] of P with respect to some facet. Now, choose k ∈ N such that 2 ≤ k ≤ d + 1. the k-shadow of H . As P is 2-neighborly we have that S 2 (H (P)) = K n and thus χ s (H (P)) = n. Obviously, S 2 (S k (H (P))) = K n and χ s (S k (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.

Embeddability
The first part of this section gives insight into the structure of neighborhoods of single vertices in a hypergraph H ∈ H d,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
We say H is piecewise linearly embeddable if there exists ψ : ϕ(H ) → R d such that ψ is a homeomorphism from ϕ(H ) onto its image and there exists a (locally finite) subdivision K of ϕ(H ) (seen as a geometric simplicial complex) such that ψ is affine on all elements of K . We call ψ a piecewise linear embedding of H into R d and we denote by H PL d,k the set of all piecewise linearly d-embeddable k-uniform hypergraphs.

Lemma 4 For a hypergraph H ∈ H PL d,k on n vertices, d ≥ k ≥ 2, and for any vertex
i. e. all points in ε · ϕ(H v ) are so close to 0 2k−1 that they lie completely in elements of K v that contain the origin.
be the set of all subdivision points of K 2 v without 0 2k−1 and let Obviously, we have that δ > 0. We take a regular d-simplex T ⊆ R d centered at the origin with sides of length δ and set C = ∂ T . Due to our choice of δ, all ψ(w) for w ∈ V K 2 v lie outside of T . Further, for all e ∈ K 2 v the intersection ψ(e) ∩ C is the union of finitely many at most (k − 2)-dimensional simplices and homeomorphic to We denote the set of subdivision points without 0 2k−1 by Finally, note that C\{x} is piecewise linearly homeomorphic to R d−1 [25, 3.20]. Let γ be such a (piecewise linear) homeomorphism . Then 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].

Lemma 5 (a) For a hypergraph H ∈ H PL
k,k on n vertices, k ≥ 2, we have that (b) For a hypergraph H ∈ H PL k+1,k+1 on n vertices, k ≥ 2, and for any vertex v we have that 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 ∈ H PL k+1,k+1 , v one of the n vertices. By Lemma 4, NH H (v) ∈ H PL k,k . By (a), Given that (b) is true for some k ≥ 2, we show that (a) holds for k + 1. Let . As e itself counts for the degree as well, one 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 [ Proof This follows from inductively applying Lemma 4 and Proposition 7. 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 = {w 1 , . . . , w m } ⊆ R d be distinct points on the moment curve in that order and P = conv W . We call a q-element subset 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 where all X i , Y S , and Y E are contiguous sets, Y S = ∅ or w 1 ∈ Y S , Y E = ∅ or w m ∈ Y E , and at most d − k sets X i have odd cardinality.
Shephard's 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 = U 1 ∪ U 2 where U 1 and U 2 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 = Y S ∪ X 1 ∪ · · · ∪ X t ∪ Y E holds where at most d − k of the contiguous sets X i have odd cardinality.
Proof The two edges U 1 and U 2 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 R 3 in arbitrary order. Then the line segments AB and C D do not intersect.
Proof This follows immediately from Corollary 13 for the case k = 2 and d = 3.
In the k = d = 3 case Corollary 13 allows zero odd sets X i . Thus, we can easily classify all possible configurations for two edges.

Lemma 15
Let H be a 3-uniform hypergraph and ϕ : V (H ) → R 3 such that ϕ 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 ϕ(e ∪ f ) on the moment curve has one of the Configurations 1-12 shown in Table 3. Then ϕ is an embedding of H . Proof Note that the relative order of edges with two common vertices is irrelevant as they always intersect according to Definition 1. Configurations 1-11 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 Assume otherwise. Note that if two triangles intersect in R 3 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{ψ(x j 1 ,0 ), ψ(x j 2 ,0 )} ∩ conv ψ(D 1 ) = ∅. However, by Theorem 12 we know that conv{ψ(x j 1 ,0 ), ψ(x j 2 ,0 )} is a face of the polytope Note that if we have two edges with vertices on the moment curve as in Configurations 13-16 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.

Bounding the Weak Chromatic Number
For d, k, n ∈ N we define 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 χ w d,k (n). Obviously, χ w d,k (n) is monotonically increasing in n and in d and monotonically decreasing in k if the other parameters remain fixed.
Remark 16 (a) For k = 2, the results in Tables 1 and 2

follow from the Four Color
Theorem and the fact that all graphs are d-embeddable for d ≥ 3.
n is (2k − 1)-embeddable for all k ∈ N by the Menger-Nöbeling Theorem (see [20, p. 295] and [22]) and χ w K In any triangle {u, v, w} of H under the coloring κ these vertices have exactly three different colors. Therefore, under the coloring κ at least one vertex with color 1 and one vertex with color 2 exists. Thus κ is a valid 2-coloring of H .

Theorem 18 Let d ≥ 3. Then one has
We want to apply the Lovász 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 for any one edge to be bad is p = 1 c d−1 . Moreover, let e be any edge in H and F be 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 Lovász 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 Choosing an integer the hypergraph H is c-colorable and χ w (H ) ≤ c.

Theorem 19 Let d ≥ ≥ 3. Then one has
This result also holds for piecewise linear embeddings.
Proof By Corollary 9 we know that every edge is adjacent to at most = (d− +2)! (d−1)! · n d−1−3 −1−d − d other edges. The rest of the proof is now analogous to the proof of Theorem 18.

Theorem 20 Let d ≥ 2. Then one has
Proof By Corollary 11 we know that every edge is adjacent to at most = dn (d−1)/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
log n 2 log log n − 1 = log n log log n as n → ∞.
Proof We first define a sequence of hypergraphs H m for m ≥ 2 such that    Relative order of i 1 , i 2 , j 1 , j 2 Case number j 1 = j 2 and i 1 = i 2 10 j 1 = j 2 and i 1 = i 2 Two shared vertices Now, arrange the vertices of H m on the moment curve in that order and pick any two edges e 1 and e 2 . By Lemma 15 we can assume that they do not share two vertices.  Table 3 and thus they intersect according to Definition 1.
Case 4: e 1 = {v i 1 , v j 1 , v} and e 2 = {v i 2 , v j 2 , w}. Again, i 1 < j 1 and i 2 < j 2 holds. Without loss of generality assume j 1 ≤ j 2 . We then have one of the cases listed in Table 5. (w). This is also shown in Fig. 2.
Arrange the vertices of H d m on the moment curve in that order and pick any two edges g 1 and g 2 .
Case 1: Both edges are from the subhypergraph H d m−1 . Then they intersect in accordance to Definition 1 and Corollary 13 as their relative order reflects that of f The last number, by assumption, was bounded from above by (d − 1) − 3 for at least one e i , i ∈ {1, 2} (unless d = 4 and they intersect according to Case 12 in Table 3, see below). So at least one g i has at most d − 3 odd contiguous subsets. Thus, the order given by f (d) m provides an embedding of H d m . Note that there is one small exception to Case 3 when d = 4. Here, e 1 and e 2 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 g 1 and g 2 at least one of the edges continues to have only one odd contiguous subset (see Table 6). 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.