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, LudwigMaximiliansUniversitt 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 kuniform 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 nvertex 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 kuniform 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 facechromatic 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 kuniform hypergraph if the vertex set V is a
finite set and the edge set E consists of kelement 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 kuniform hypergraph of
order n.
Let H be a kuniform hypergraph. A function : V (H ) {1, . . . , c} is said to
be a weak ccoloring if for all e E (H ) the property (e) > 1 holds. The function
is said to be a strong ccoloring 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 (dEmbeddings) Let H be a kuniform 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 dembeddable
kuniform 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 nonindependent 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 kuniform hypergraph H is said to be dembeddable
if there exists an embedding of H into Rd . Also, we denote by Hd,k the set of all
dembeddable kuniform hypergraphs.
One can easily see that our definition of 2embeddability 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
kuniform hypergraph is dembeddable is decidable and in PSPACE [23]. One can
show that the aforementioned three types of embeddings are equivalent only in the
less than 3dimensional 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 dembeddable
kuniform hypergraph on n vertices. All results which only follow nontrivially 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 dembeddable
kuniform 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 2neighborly [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 kshadow of H . As P is 2neighborly 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 kuniform
hypergraph on n vertices that is linearly dembeddable 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 ddimensional 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 dembeddings) Let H be a kuniform 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 dembeddable kuniform
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 kuniform 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 subhypergraph 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 dembeddable 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 kuniform
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 kuniform
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 qelement 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 kuniform 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 3uniform hypergraph and : V (H ) R3 such that maps
all vertices onetoone 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 dembeddable kuniform 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 4coloring 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 2coloring 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 ccoloring of the vertices of H an edge is called bad if
it is monochromatic and good if not. In a uniformly random ccoloring 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 ccolorable. 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 ccolorable 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 Subcases of Case 3 in
the proof of Theorem 21
referring to the corresponding
cases of Lemma 15
Table 5 Subcases 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
2colorable. 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 logarithmicpolynomial
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 993/101. 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 309/22.
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