# Lower bounds for measurable chromatic numbers

###### Abstract.

The Lovász theta function provides a lower bound for the chromatic number of finite graphs based on the solution of a semidefinite program. In this paper we generalize it so that it gives a lower bound for the measurable chromatic number of distance graphs on compact metric spaces.

In particular we consider distance graphs on the unit sphere. There we transform the original infinite semidefinite program into an infinite linear program which then turns out to be an extremal question about Jacobi polynomials which we solve explicitly in the limit. As an application we derive new lower bounds for the measurable chromatic number of the Euclidean space in dimensions and we give a new proof that it grows exponentially with the dimension.

###### Key words and phrases:

Nelson-Hadwiger problem, measurable chromatic number, semidefinite programming, orthogonal polynomials, spherical codes###### 1991 Mathematics Subject Classification:

52C10, 52C17, 90C22## 1. Introduction

The chromatic number of the -dimensional Euclidean space is the minimum number of colors needed to color each point of in such a way that points at distance from each other receive different colors. It is the chromatic number of the graph with vertex set and in which two vertices are adjacent if their distance is . We denote it by .

A famous open question is to determine the chromatic number of the plane. In this case, it is only known that , where lower and upper bounds come from simple geometric constructions. In this form the problem was considered, e.g., by Nelson, Isbell, Erdős, and Hadwiger. For historical remarks and for the best known bounds in other dimensions we refer to Székely’s survey article [21]. The first exponential asymptotic lower bound is due to Frankl and Wilson [8, Theorem 3]. Currently the best known asymptotic lower bound is due to Raigorodskii [17] and the best known asymptotic upper bound is due to Larman and Rogers [12]:

In this paper we study a variant of the chromatic number of , namely the measurable chromatic number. The measurable chromatic number of is the smallest number such that can be partitioned into Lebesgue measurable stable sets. Here we call a set stable if no two points in lie at distance from each other. In other words, we impose that the sets of points having the same color have to be measurable. We denote the measurable chromatic number of by . One reason to study the measurable chromatic number is that then stronger analytic tools are available.

The study of the measurable chromatic number started with Falconer [7], who proved that . The measurable chromatic number is at least the chromatic number and it is amusing to notice that in case of strict inequality the construction of an optimal coloring necessarily uses the axiom of choice.

Related to the chromatic number of the Euclidean space is the chromatic number of the unit sphere . For , we consider the graph whose vertices are the points of and in which two points are adjacent if their inner product equals . The chromatic number of and its measurable version, denoted by and respectively, are defined as in the Euclidean case.

The chromatic number of this graph was studied by Lovász [14], in particular in the case when is small. He showed that

Frankl and Wilson [8, Theorem 6] showed that

The (measurable) chromatic number of provides a lower bound for the one of : After appropriate scaling, every proper coloring of intersected with the unit sphere gives a proper coloring of the graph , and measurability is preserved by the intersection.

In this paper we present a lower bound for the measurable chromatic number of . As an application we derive new lower bounds for the measurable chromatic number of the Euclidean space in dimensions and we give a new proof that it grows exponentially with the dimension.

The lower bound is based on a generalization of the Lovász theta function (Lovász [13]), which gives an upper bound to the stability number of a finite graph. Here we aim at generalizing the theta function to distance graphs in compact metric spaces. These are graphs defined on all points of the metric space where the adjacency relation only depends on the distance.

The paper is structured as follows: In Section 2 we define the stability number and the fractional measurable chromatic number and give a basic inequality involving them. Then, after reviewing Lovász’ original formulation of the theta function in Section 3, we give our generalization in Section 4. Like the original theta function for finite graphs, it gives an upper bound for the stability number. Moreover, in the case of the unit sphere, it can be explicitly computed, thanks to classical results on spherical harmonics. The material needed for spherical harmonics is given in Section 5 and an explicit formulation for the theta function of is given in Section 6.

In Section 7 we choose specific values of for which we can analytically compute the theta function of . This allows us to compute the limit of the theta function for the graph as goes to in Section 8. This gives improvements on the best known lower bounds for in several dimensions. Furthermore this gives a new proof of the fact that grows exponentially with . Although this is an immediate consequence of the result of Frankl and Wilson (and of Raigorodskii, and also of a result of Frankl and Rödl [9]) and our bound of is not an improvement, our result is an easy consequence of the methods we present. Moreover, we think that our proof is of interest because the methods we use here are radically different from those used before. In particular, they can be applied to other metric spaces.

In Section 9 we point out how to apply our generalization to distance graphs in other compact metric spaces, endowed with the continuous action of a compact group. Finally in Section 10 we conclude by showing the relation between our generalization of the theta function and the theta function for finite graphs of and by showing the relation between our generalization and the linear programming bound for spherical codes established by Delsarte, Goethals, and Seidel [6].

## 2. The fractional chromatic number and the stability number

Let be a finite or infinite graph whose vertex set is equipped with the measure . We assume that the measure of is finite. In this section we define the stability number and the measurable fractional chromatic number of and derive the basic inequality between these two invariants. In the case of a finite graph one recovers the classical notions if one uses the uniform measure for .

Let be the Hilbert space of real-valued square-integrable functions defined over with inner product

for . The constant function is measurable and its squared norm is the number . The characteristic function of a subset of we denote by .

A subset of is called a measurable stable set if is a measurable set and if no two vertices in are adjacent. The stability number of is

Similar measure-theoretical notions of the stability number have been considered before by other authors for the case in which is the Euclidean space or the sphere . We refer the reader to the survey paper of Székely [21] for more information and further references.

The fractional measurable chromatic number of is denoted by . It is the infimum of where and are nonnegative real numbers such that there exist measurable stable sets satisfying

Note that the measurable fractional chromatic number of the graph is a lower bound for its measurable chromatic number.

###### Proposition 2.1.

We have the following basic inequality between the stability number and the measurable fractional chromatic number of a graph :

(1) |

So, any upper bound for provides a lower bound for .

###### Proof.

Let be nonnegative real numbers and be measurable stable sets such that . Since is measurable, its characteristic function lies in . Hence

## 3. The Lovász theta function for finite graphs

In the celebrated paper [13] Lovász introduced the theta function for finite graphs. It is an upper bound for the stability number which one can efficiently compute using semidefinite programming. In this section we review its definition and properties, which we generalize in Section 4.

The theta function of a graph is defined by

(2) |

###### Theorem 3.1.

For any finite graph , .

Although this result follows from [13, Lemma 3] and [13, Theorem 4], we give a proof here to stress the analogy between the finite case and the more general case we consider in our generalization Theorem 4.1.

###### Proof of Theorem 3.1.

Let be a stable set. Consider the characteristic function of and define the matrix by

Notice satisfies the conditions in (2). Moreover, we have , and so . ∎

###### Remark 3.2.

If the graph has a nontrivial automorphism group, it is not difficult to see that one can restrict oneself in (2) to the functions which are invariant under the action of any subgroup of , where is the automorphism group of , i.e., it is the group of all permutations of that preserve adjacency. Here we say that is invariant under if holds for all and all . If moreover acts transitively on , the second condition is equivalent to for all .

## 4. A generalization of the Lovász theta function for distance graphs on compact metric spaces

We assume that is a compact metric space with distance function . We moreover assume that is equipped with a nonnegative, Borel regular measure for which is finite. Let be a closed subset of the image of . We define the graph to be the graph with vertex set and edge set .

The elements of the space consisting of all continuous functions are called continuous Hilbert-Schmidt kernels; or kernels for short. In the following we only consider symmetric kernels, i.e., kernels with for all . A kernel is called positive if, for any nonnegative integer , any points , and any real numbers , we have

We are now ready to extend the definition (2) of the Lovász theta function to the graph . We define

(3) |

###### Theorem 4.1.

The theta function is an upper bound for the stability number, i.e.,

###### Proof.

Fix arbitrarily. Let be a stable set such that . Since is regular, we may assume that is closed, as otherwise we could find a stable set with measure closer to and use a suitable inner-approximation of it by a closed set.

Note that, since is compact and stable, there must exist a number such that for all and . But then, for small enough , the set

where is the distance from to the closed set , is stable. Moreover, notice that is open and that, since it is stable, .

Now, the function given by

for all is continuous and such that and . So the kernel given by

for all is feasible in (3).

Let us estimate the objective value of . Since we have

and

we finally have

and, since is arbitrary, the theorem follows. ∎

Let us now assume that a compact group acts continuously on , preserving the distance . Then, if is a feasible solution for (3), so is for all . Averaging on leads to a -invariant feasible solution

where denotes the Haar measure on normalized so that has volume . Moreover, observe that the objective value of is the same as that of . Hence we can restrict ourselves in (3) to -invariant kernels. If moreover is homogeneous under the action of , the second condition in (3) may be replaced by for all .

We are mostly interested in the case in which is the unit sphere endowed with the Euclidean metric of , and in which is a singleton. If and , so that if and only if , the graph is denoted by . Since the unit sphere is homogeneous under the action of the orthogonal group , the previous remarks apply.

## 5. Harmonic analysis on the unit sphere

It turns out that the continuous positive Hilbert-Schmidt kernels on the sphere have a nice description coming from classical results of harmonic analysis reviewed in this section. This allows for the calculation of . For information on spherical harmonics we refer to [1, Chapter 9] and [23].

The unit sphere is homogeneous under the action of the orthogonal group , where denotes the identity matrix. Moreover, it is two-point homogeneous, meaning that the orbits of on pairs of points are characterized by the value of their inner product. The orthogonal group acts on by , and is equipped with the standard -invariant inner product

(4) |

for the standard surface measure . The surface area of the unit sphere is .

It is a well-known fact (see e.g. [23, Chapter 9.2]) that the Hilbert space decomposes under the action of into orthogonal subspaces

(5) |

where is isomorphic to the -irreducible space

of harmonic polynomials in variables which are homogeneous and have degree . We set . The equality in (5) means that every can be uniquely written in the form , where , and where the convergence is in the -norm.

The addition formula (see e.g. [1, Chapter 9.6]) plays a central role in the characterization of -invariant kernels: For any orthonormal basis of and for any pair of points we have

(6) |

where is the normalized Jacobi polynomial of degree with parameters , with and . The Jacobi polynomials with parameters are orthogonal polynomials for the weight function on the interval . We denote by the normalized Jacobi polynomial of degree with normalization .

In [18, Theorem 1] Schoenberg gave a characterization of the continuous kernels which are positive and -invariant: They are those which lie in the cone spanned by the kernels . More precisely, a continuous kernel is -invariant and positive if and only if there exist nonnegative real numbers such that can be written as

(7) |

where the convergence is absolute and uniform.

## 6. The theta function of

We obtain from Section 4 in the case , , and , the following characterization of the theta function of the graph :

(8) |

(It will be clear later that the maximum above indeed exists.)

###### Corollary 6.1.

We have

A result of de Bruijn and Erdős [4] implies that the chromatic number of is attained by a finite subgraph of it. So one might wonder if computing the theta function for a finite subgraph of could give a better bound than the previous corollary. This is not the case as we will show in Section 10.

The theta function for finite graphs has the important property that it can be computed in polynomial time, in the sense that it can be approximated with arbitrary precision using semidefinite programming. We now turn to the problem of computing the generalization (8).

First, we apply Schoenberg’s characterization (7) of the continuous kernels which are -invariant and positive. This transforms the original formulation (3), which is a semidefinite programming problem in infinitely many variables having infinitely many constraints, into the following linear programming problem with optimization variables :

(9) |

where .

To obtain (9) we simplified the objective function in the following way. Because of the orthogonal decomposition (5) and because the subspace contains only the constant functions, we have

We furthermore used and .

###### Theorem 6.2.

Let be the minimum of for Then the optimal value of (9) is equal to

###### Proof.

We first claim that the minimum exists and is negative. Indeed, if for all , then (9) either has no solution (in the case that all are positive) or in any solution, which contradicts Theorem 4.1. So we know that for some , . This, combined with the fact that goes to zero as goes to infinity (cf. [1, Chapter 6.6] or [20, Chapter 8.22]), proves the claim.

Let be so that . It is easy to see that there is an optimal solution of (9) in which only and are positive. Hence, solving the resulting system

gives and and the theorem follows. ∎

###### Example 6.3.

The minimum of for is attained at . It is a rational number and its first decimal digits are .

## 7. Analytic solutions

In this section we compute the value

for specific values of . Namely we choose to be the largest zero of an appropriate Jacobi polynomial.

Key for the discussion to follow is the interlacing property of the zeroes of orthogonal polynomials. It says (cf. [20, Theorem 3.3.2]) that between any pair of consecutive zeroes of there is exactly one zero of .

We denote the zeros of by with and with the increasing ordering . We shall need the following collection of identities:

(10) |

(11) | ||||

(12) | ||||

(13) | ||||

(14) | ||||

(15) | ||||

(16) |

They can all be found in [1, Chapter 6], although with different normalization. Formula (10) is [1, (6.3.9)]; (11) and (12) are [1, (6.4.23)]; (13) is [1, (6.3.8)], (14) is [1, (6.4.21)]; (15) follows by the change of variables from (14) and (11), (12); (16) is [1, (6.4.20)].

###### Proposition 7.1.

Let be the largest zero of the Jacobi polynomial . Then, .

###### Proof.

We start with the following crucial observation: From (13), is a zero of the derivative of . Hence it is a minimum of because it is the last extremal value in the interval and because , whence (using (13)) is increasing on .

Now we prove that for all where we treat the cases and separately.

It turns out that the sequence is decreasing for . From (16), the sign of equals the sign of . We have the inequalities

The first one is a consequence of the interlacing property. From (15) one can deduce that has exactly one zero in the interval since it changes sign at the extreme points of it, and by the same argument has a zero left to . Thus, . So lies to the right of the largest zero of and hence which shows that for .

Let us consider the case . The inequality [1, (6.4.19)] implies that

(17) |

The next observation, which finishes the proof of the lemma, is stated in [1, (6.4.24)] only for the case :

(18) |

To prove it consider

Applying (10) in the computation of shows that

The polynomial takes positive values on and hence is increasing on this interval. In particular,

which simplifies to

Since are the local extrema of , we have proved (18). ∎

## 8. New lower bounds for the Euclidean space

In this section we give new lower bounds for the measurable chromatic number of the Euclidean space for dimensions . This improves on the previous best known lower bounds due to Székely and Wormald [22]. Table 8.1 compares the values. Furthermore we give a new proof that the measurable chromatic number grows exponentially with the dimension.

For this we give a closed expression for which involves the Bessel function of the first kind of order (see e.g. [1, Chapter 4]). The appearance of Bessel functions here is due to the fact that the largest zero of the Jacobi polynomial behaves like the first positive zero of the Bessel function . More precisely, it is known [1, Theorem 4.14.1] that, for the largest zero of the polynomial ,

(19) |

and, with our normalization (cf. [1, Theorem 4.11.6]),

(20) |

###### Theorem 8.1.

We have

###### Proof.

First we show that

(21) |

We estimate the difference

that we upper bound by

The second term tends to from (20). Define by . By the mean value theorem we have

where denotes the interval with extremes and . Then, with (19),

and for all

From (13),

Hence we have proved that

and (21) follows.

Since the zeros tend to as tends to infinity, to prove the theorem it suffices to show that exists. This follows from (21) and the following two facts which hold for all :

(22) |

and

(23) |

Fact (22) follows from (18) and [1, (6.4.19)]. For establishing fact (23) we argue as follows: As in the proof of Proposition 7.1, we use (15) to show that has exactly one zero in the interval , namely . From (16) we then see that is the only point in this interval where and coincide. Now it follows from the interlacing property that is increasing in the interval and that is decreasing in the interval, and we are done. ∎

###### Corollary 8.2.

We have

where .∎

We use this corollary to derive new lower bounds for . We give them in Table 8.1. For our bounds are worse than the existing ones and for our bound is which is also the best known one.

In fact Oliveira and Vallentin [16] show, by different methods, that the above bound is actually a bound for . This then gives improved bounds starting from . With the use of additional geometric arguments one can also get a new bound for in this framework.

best lower bound | new lower bound | |

previously known for | for | |

10 | 45 | 48 |

11 | 56 | 64 |

12 | 70 | 85 |

13 | 84 | 113 |

14 | 102 | 147 |

15 | 119 | 191 |

16 | 148 | 248 |

17 | 174 | 319 |

18 | 194 | 408 |

19 | 263 | 521 |

20 | 315 | 662 |

21 | 374 | 839 |

22 | 526 | 1060 |

23 | 754 | 1336 |

24 | 933 | 1679 |