Coloring d-Embeddable k-Uniform Hypergraphs

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


Introduction
The Four Color Theorem [AH77,AHK77] 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 [Hea90], Ringel and Youngs [RY68], and many others.
In this paper, we consider graphs and hypergraphs that are embeddable into d for d ≥ 3 in such a way that their edges do not intersect (see Definition 1 below). For graphs, however, this is not a very interesting question because for any n ∈ the vertices of the complete graph K n can be embedded into 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. E. g., this follows trivially from Corollary 11 in the case of k = 2 and d = 3.
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. [Men28,MTW11,vK33,Flo34]), but to our surprise, we have not been able to find any bounds on their chromatic number.
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 ⊆ We next define what we mean when we say that a hypergraph is embedabble into d . 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 ∈ . A (linear) embedding of H into d is a function ϕ : V (H) → d , where ϕ(A) for A ⊆ V (H) is to be interpreted pointwise, such that • dim aff ϕ(e) = k − 1 for all e ∈ E(H) and • conv ϕ e 1 ∩ e 2 = conv ϕ(e 1 ) ∩ conv ϕ(e 2 ) for all e 1 , e 2 ∈ E(H) 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 d . Also, we denote by 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 [Fár48]. 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 Section 1 in [MTW11]. 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 [Ren92]. The aforementioned three types of embeddings have shown to be equivalent only in the less than 3-dimensional case (see e. g. [Bre83,BG00]). 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.
We can now give a summary of our main results in the following Tables 1.1, 1.2, and 1.3, which contain upper or lower bounds for the maximum weak respectively strong 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. To conclude the introduction here is a rough outline for the rest of the paper. In Section 2 the general concept of embedding hypergraphs into d-dimensional space is discussed and 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 ). Section 3 then applies this to the strong coloring problem of hypergraphs and, finally, Section 4 presents our current level of knowledge for the apparently 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 ∈ d,k . We will later use them 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.

Definition 3 (Neighborhoods)
For a k-uniform hypergraph H and a vertex v ∈ V (H) we say the neighborhood of v is N H (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 the induced (k − 1)-uniform hypergraph

Lemma 4
For a hypergraph H ∈ PL d,k on n vertices, d ≥ k ≥ 2, and for any vertex v we have that Then there exists an > 0 such that i. e. all points in · ϕ(H v ) are so close to 0 2k−1 that they lie completely in elements of K v that contain the origin.
This is true as each simplex ψ(e), e ∈ K 2 v , has dimension less or equal k − 1 and hence "forbids" only a null set of possible lines through the origin when d ≥ k. Without loss of generality, let S = × {0 d−1 }.
and π 1 : d → be the natural projections onto the last d − 1 coordinates and the first coordinate respectively. Further, put δ = min{ π d−1 (ψ(w)) : w ∈ V K 2 v } > 0 and ∆ = max{|π 1 (ψ(w))| : We take a regular (d −1)-simplex T ⊆ d−1 centered at the origin with sides of length δ and set . Thus, C is the boundary of a stretched dsimplex. Due to our choice of δ and ∆, all ψ(w) for w ∈ V K 2 v lie outside of C and for all e ∈ K 2 v the intersection ψ(e) ∩ C completely lies in C := C\({−2∆} × T ), is connected, and the union of finitely many at most (k − 2)-dimensional simplices .
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 ∈ PL k+1,k+1 , v one of the n vertices. By Lemma 4, Given that (b) is true for some k ≥ 2, we show that (a) holds for k

Corollary 6
For a hypergraph H ∈ PL k,k on n vertices, k ≥ 3, and for any edge e ∈ E(H) there exist at most Proof. This follows from Lemma 5, since every edge has exactly k vertices and each of them has at most degree 6n k−2 −12n k−3 (k−1)! . As e itself counts for the degree as well, one can subtract 1.
We need to bound the number of edges in a d-embeddable hypergraph to prove upper bounds for the chromatic number. The following proposition will also help to do this. Note that there exist much stronger conjectured bounds (see [Kal02, Conjecture 27] and [Gun09, Conjecture 1.4.4]).

Corollary 8
Using Lemma 4 and Proposition 7, we inductively obtain that for H ∈ PL 2k−l,k on n vertices,

Corollary 9
For a hypergraph H ∈ PL 2k−l,k on n vertices, k ≥ l ≥ 3, and for any edge e ∈ E(H) there exist at most kn k−1−3 l−1−k − 1 other edges adjacent to it.
Proof. This fact follows analogously to Corollary 6 from Corollary 8.
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 10 (Shephard [She68])
Let W = {w 1 , . . . , w m } ⊆ d be distinct points on the moment curve in that order and P = conv W . We say a q-element subset 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 11
In the setting of Theorem 10 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 at least one of them is a face of P = conv W . It thus suffices to show that for one j ∈ {1, 2} holds where at most d − k of the contiguous sets X i have odd cardinality.
In the k = d = 3 case Corollary 11 allows zero odd sets X i . Thus, we can easily classify all possible configurations for two edges.

Corollary 12
Given a 3-uniform hypergraph H and ϕ : V (H) → 3 . Then ϕ is an embedding of H if 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 Configurations 1-10 shown in Figure 2.1. The relative order of edges with two common vertices is irrelevant. Proof. Having in mind Corollary 12, it is sufficient to prove the following: Assume otherwise. Note that if two triangles intersect in 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 10 we know that conv{ψ(x j 1 ,0 ), ψ(x j 2 ,0 )} is a face of the polytope P = conv({ψ(x j 1 ,0 ), ψ(x j 2 ,0 )} ∪ ψ(D 1 )) which is a contradiction.

Strong colorings
For d, k, n ∈ we define to be the maximum strong chromatic number of a d-embeddable k-uniform hypergraph on n vertices.
Clearly, χ s d,k (n) is monotonically increasing in n and in d and monotonically decreasing in k if the other parameters remain fixed. Furthermore, it is not difficult to establish some kind of strict simultaneous monotonicity as follows:
H) and consider the embedding Then, H = (V , E ) is in d+1,k+1 using the embedding ϕ . Assume that H has a strong coloring with at most c colors. By the construction of E , this would yield a coloring with less than c colors for H, which contradicts the choice of c.
We now give lower bounds on χ s d,k (n), which we essentially derive by retreating to graphs.

Definition 15 (The shadow of a hypergraph)
Let H be a k-uniform hypergraph, k ≥ 2. Then we call, following For a k-uniform hypergraph H, we have where χ(G) is the classical chromatic number of a graph G.

Remark 16
a) The Four Color Theorem and Equation (2) imply that χ s 2,k (n) ≤ 4 for k ∈ {2, 3}. Obviously, there are graphs and hypergraphs for which this bound is sharp. b) For d ≥ 2k − 1, we have χ s d,k (n) = n as K (k) n is (2k − 1)-embeddable for all k ∈ by the Menger-Nöbeling Theorem (see [Men28,p. 295] and [Nöb31]) and χ s K (k) n = n. c) For d ≤ k − 2, we know χ s d,k (n) = 1 as H ∈ d,k cannot have any edge.
Proof. It is sufficient that for m ≥ 4 we can find a hypergraph H ∈ 3,4 on m 2 vertices with strong chromatic number larger or equal m. Let m ≥ 4 and let G = K m be the complete graph on m vertices. We can use the embedding ϕ from Equation (1) to embed G into 3 .
Note that G has only finitely many edges. So for every embedded edge e = {u, v} of G there exists a small open convex set C e ⊆ 3 such that (conv ϕ(e))\ϕ(e) ⊆ C e , ϕ(e) ⊆ ∂ C e , and Then ϕ is an embedding of H as each edge forms a tetrahedron inside its corresponding set C e and intersects with other tetrahedrons at most at its two endpoints. Hence, H ∈ 3,4 . Obviously, |V | = m 2 and χ s (H) = χ( (H)) ≥ χ(G) = χ(K m ) = m.

Corollary 18
Together with Lemma 14 we obtain for large n and d ≥ 3 that Using monotonicity the same holds for all d ≥ 3, k ≤ d + 1 and χ s d,k (n).
It is now left to show that F m is 3-embeddable for all m. Define ϕ(x) = x, x 2 , x 3 for x ∈ . Set ψ m (v 0 ) = ϕ(1), ψ m (v i ) = ϕ(2i), and ψ m (w i ) = ϕ(2i + 1) for all i ∈ {1, . . . , m}. ψ m is an embedding of F m : We use induction on m and for m = 0 or m = 1 this is trivial. Now assume that this is true for some m. Take any two edges e 1 = e 2 ∈ E m+1 . Observe that ψ m+1 |V m = ψ m . If e 1 , e 2 ∈ E m , then they intersect according to Definition 1 because ψ m was an embedding of F m . If both edges are in E m+1 \E m then they share the two vertices v m+1 and w m+1 . Thus, they also intersect according to Definition 1.
When e 1 ∈ E m+1 \E m and e 2 ∈ E m , we have to distinguish the several different cases of Corollary 12. If e 1 and e 2 share one vertex, we have one of Cases 6-8 and we are done. So, assume e 1 and e 2 to be disjoint. Let 0 ≤ i, j, k ≤ m be pairwise distinct. If e 1 = {v i , v m+1 , w m+1 } and e 2 = {w i , v k , w k }, then i < k and we have Case 3. If e 1 = {w i , v m+1 , w m+1 } and e 2 = {v i , v k , w k }, we have Case 2. The last possibility is that Now, let n be even, n = 2m. Note that by monotonicity we already have χ s 3,3 (2m) ≥ χ s 3,3 (2m − 1) ≥ 2m − 1. Take F m−1 and add one more vertex to obtain

Corollary 20
Together with Lemma 14 we obtain that By monotonicity, χ s d,k (n) = n holds for all d ≥ 3, k ≤ d.
Thus, except for the cases where k = d + 1, the maximum strong coloring problem was solved. In particular, we have shown that an unbounded number of colors can be necessary for any strong coloring of a d-embeddable hypergraph if d > 2.

Weak colorings
For d, k, 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. Note that an equivalent of Lemma 14 is not true for weak colorings as the existence of one vertex incident with all edges automatically implies the existence of a weak 2-coloring. 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 23
Let d ≥ 3. Then one has This result also holds for piecewise linear embeddings.
By Corollary 6 we know that every edge is at most adjacent to We want to apply the Lovász Local Lemma [EL75,Spe77] to bound the weak chromatic number of H. Let c ∈ . In any c-coloring of the vertices of H an edge is called bad if it is monochromatic and good if not. In any uniformly random c-coloring the probability for one edge to be bad is p = 1 c d−1 . Moreover, note that the events whether one edge and a set of edges not adjacent to the first edge are bad are independent. Thus each event whether an edge is bad depends on at most ∆ other such events.
The Lovász Local Lemma guarantees us that there is positive probability that 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 24
Let d ≥ l ≥ 3. Then one has This result also holds for piecewise linear embeddings.
Proof. By Corollary 9 we know that every edge is at most adjacent to ∆ = d n d−1−3 l−1−d − 1 other edges. The rest of the proof is now analogous to the proof of Theorem 23.
By monotonicity, the upper bounds presented here also hold if the uniformity of the hypergraph is larger than stated in Theorems 23 and 24. In the remaining part of this section, we now consider lower bounds for the chromatic number of hypergraphs.

Theorem 25
As n → ∞ one has    Table 4.1.

Case number
Relative order of i 1 , i 2 , j 1 , j 2 i 1 = i 2 and j 1 = j 2 10 i 2 = i 1 and j 1 = j 2 two shared vertices i 1 = i 2 and j 1 = j 2    . We then use Corollary 11 to prove that H d m is embeddable 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 11 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. Finally, we look at the case g 1 = ({v 1 } ∪ e 1 ) and g 2 = ({v 2 } ∪ e 2 ). Then their vertex sets have at most one more odd contiguous subset than the edges e 1 and e 2 had in the ordering of f d−1 m . The last number, by assumption, was bounded from above by (d − 1) − 3 for at least on e i , i ∈ {1, 2}. 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 when d = 4. Here, e 1 and e 2 could be in the relative position of Case 11 in Figure 2.1 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 Figure 4.4). Note that by monotonicity also χ w 2d−2,d (n) = Ω log n log log n .