Axiom of choice and chromatic number of R n

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1 Journal of Combinatorial Theory, Series A 110 (2005) Note Axiom of choice and chromatic number of R n Alexander Soifer a,b,1 a Princeton University, Mathematics, DIMACS, Rutgers University, Piscataway NJ, USA b University of Colorado at Colorado Springs, 1420 Austin Bluffs Parkway, Colorado Springs, CO 80918, USA Received 5 August 2004 Communicated by Victor Klee I know of mathematicians who hold that the axiom of choice has the same character of intuitive self-evidence that belongs to the most elementary laws of logic on which mathematics depends. It has never seemed so to me. Alonzo Churchs 2 Abstract In previous papers (J. Combin Theory Ser. A 103 (2003) 387) and (J. Combin. Theory Ser. A 105 (2004) 359) Saharon Shelah and I formulated a conditional chromatic number theorem, which described a setting in which the chromatic number of the plane takes on two different values depending upon the axioms for set theory. We also constructed examples of a distance graph on the real line R and difference graphs on the real plane R 2 whose chromatic numbers depend upon the system of axioms we choose for set theory. Ideas developed there are extended in the present paper to construct difference graphs on the real space R n, whose chromatic number is a positive integer in the Zermelo Fraenkelchoice system of axioms, and is not countable (if it exists) in a consistent system of axioms with limited choice, studied by Solovay (Ann. Math. Ser. 2 (1970) 1). These examples illuminate how heavily combinatorial results can depend upon the underlying set theory, help appreciate the potential complexity of the chromatic number of n-space problem, and suggest that the chromatic number of n-space may depend upon the system of axioms chosen for set theory Elsevier Inc. All rights reserved. Keywords: Chromatic number of the plane and space; Axiom of choice; Coloring; Combinatorial geometry; Discrete geometry; Axioms of set theory address: asoifer@uccs.edu 1 Thanks to DIMACS for a Long Term Visitor appointment and Princeton University for a Visiting Fellowship. 2 Talk at the International Congress of Mathematicians in Moscow, 1966 [Chu] /$ - see front matter 2004 Elsevier Inc. All rights reserved. doi: /j.jcta

2 170 Note / Journal of Combinatorial Theory, Series A 110 (2005) Question Define a graph U n on the set of all points of R n as its vertex set, with two points adjacent iff they are distance 1 apart. The graph U n ought to be called the unit distance n-space,and its chromatic number χ(r n ) is called the chromatic number of the Euclidean n-space. 3 Finite subgraphs of U n are called finite unit distance graphs. In 1950 the 18-year old Edward Nelson posed the problem of finding χ(r 2 ) (see the problem s history in [Soi1]). A number of relevant results were obtained under additional restrictionsonmonochromatic sets (see surveysinthefine problem monographs [KW] and [CFG], andalsoin[soi2]). Amazingly, though, the problem has withstood all assaults in the general case, leaving us with an embarrassingly wide range for χ(r 2 ) being4,5,6or7. In the early 1960s Paul Erdös generalized the chromatic number problem to n-dimensional Euclidean spaces R n, and the search for χ(r n ) began. Asymptotic exponential upper and lower bounds for χ(r n ) were found. The greatest progress was recently achieved for 3- dimensional space: 6 χ(r 3 ) 15, where the lower bound is due to Nechushtan [Nec] and the upper bound due to Coulson [Cou]. In their fundamental 1951 paper [EB], Erdös and de Bruijn showed that the chromatic number of the plane is attained on some finite subgraph. This result naturally channeled much of research in the direction of finite unit-distance graphs. One limitation of the Erdös de Bruijn result, however, has been that they used quite essentially the axiom of choice. What happens if we have no choice? This question was addressed in [SS1,SS2]. We have formulated there aconditional chromatic number theorem, which specifically described a setting in which the chromatic number of the plane takes on two different values depending upon the axioms for set theory. We have also constructed an example of a distance graph on the real line R and examples of graphs on the real plane R 2 whose chromatic number depended upon the system of axioms chosen for set theory. Ideas developed there are extended in the present paper to construct graphs on R n (as their vertex set), whose chromatic number is finite in the Zermelo Fraenkel Choice system of axioms, and is not countable (if it exists) in a consistent system of axioms with limited choice, studied by Solovay [Sol]. These graphs can be called difference graphs in the sense that the adjacency of two points is determined by their vector difference. Of course, in the case of the real line, this notion of a difference graph coincides with the notion of a distance graph. Existence of such examples suggests that likely the chromatic number of n-space depends upon the system of axioms we choose for set theory. 2. Preliminaries Basic set-theoretic definitions are included in [SS1,SS2]. Here I will repeat only the notations: 3 The chromatic number χ(g) of a graph G is the smallest number of colors required for coloring the vertices, so that no two vertices of the same color are connected by an edge.

3 Note / Journal of Combinatorial Theory, Series A 110 (2005) Axiom of choice will be denoted by AC; countable axiom of choice by AC ℵ0 ; principle of dependent choices by DC. LM will stand for Every set of real numbers is Lebesgue measurable. As always, ZF stands for the Zermelo Fraenkel system of axioms for sets, and ZFC for Zermelo Fraenkel with the addition of the axiom of choice. Assuming the existence of an inaccessible cardinal, Solovay constructed in 1964 (and published in 1970) a model that proved the following major consistency result [Sol]: Solovay Theorem. The system of axioms ZF + DC + LM is consistent. 3. Example in R n Let Z and Q denote the set of all integers and the set of all rational numbers respectively, so that Z n is the set of integral n-tuples and Q n is the rational n-space. We define a graph G as follows: the set R n of points of the n-space serves as the vertex set, and the set of edges is ni=1 {(s, t) : s, t, R n ; s t 2ε i Q n } where ε i are the n unit vectors on coordinate axes forming the standard basis of R n. For example, ε 1 = (1, 0,...,0) we will use this vector in the proof of Claim 2 below. Claim 1. In ZFC the chromatic number of the graph G is equal to 2. Proof. Let S ={q + m 2 : q Q n,m Z n }. We define an equivalence relation E on R n as follows: set s t S. Let Y be a set of representatives for E. Fort R n let y(t) Y be such a representative that tey(t).wedefine a 2-coloring c( ) as follows: c(t) = k mod2 iff there is k Z n such that t y(t) 2k Q n, where k denotes the sum of all n coordinates of k. Without AC the chromatic situation changes: Claim 2. In ZF + AC ℵ0 + LM the chromatic number of the graph G cannot be equal to any positive integer n nor even to ℵ 0. The proof of Claim 2 immediately follows from the first of the following two statements: Statement 1. If A 1,A 2,...,A k,...are measurable subsets of R n and A k =[0, 1) n, then at least one set A k contains two adjacent vertices of the graph G. Statement 2. If A [0, 1) n and A contains no pair of adjacent vertices of G, then A is null (of Lebesgue measure zero). Proof of Claim 2. We start with the proof of statement 2. Assume to the contrary that A [0, 1) n contains no pair of adjacent vertices of G 2,yetA has positive measure. Then there is an n-dimensional parallelepiped I, with a edge parallel to the first coordinate axis of length, say, a, such that μ(a I) μ(i) k<ω > (0.1)

4 172 Note / Journal of Combinatorial Theory, Series A 110 (2005) Choose q Q such that 2 <q< a.define a translate B of A as follows: B = A (q 2)ε 1, where ε 1 = (1, 0,...,0) is the unit vector on the first coordinate axis of R n. Then μ(b I) μ(i) > (0.2) Inequalities (0.1) and (0.2) imply that there is ν I A B.Sinceν B, wehave w = ν + (q 2)ε 1 A. So, we have ν,w A and ν w 2ε 1 = qε 1 Q n. Thus, {ν,w} is an edge of the graph G with both endpoints in A, which is the desired contradiction. The proof of the statement 1 is now obvious. Since k<ω A k =[0, 1) n and Lebesgue measure is a countably-additive function in AC ℵ0, there is a positive integer k such that A k is a non-null set. By statement 2, A k contains a pair of adjacent vertices of the graph G as required. 4. More examples in R n We can define a graph G as follows: the set R n of points of the n-space still serves as the vertex set, but the set of edges is 0 i =j n {(s, t) : s, t Rn ; s t 2(ε i ε j ) Q n } where ε i are the n unit vectors on coordinate axes forming a standard basis of R n,and ε 0 = 0 R n. Claim 1. In ZFC the chromatic number of the graph G is equal to 2 n. Proof. Indeed, the 2 n vertices of the n-dimensional unit cube generated by ε i, 0 i n must all be colored in different colors, so 2 n colors are obviously needed. Let Y be a set of representatives for E. Fort R n let y(t) Y be a representative such that tey(t).wedefinea2 n -coloring c(t) as follows: c(t) = (kmod2 1,k2 mod2,...,kn mod2 )iff there is k = (k 1,k 2,...,k n ) Z n such that t y(t) 2k Q n, where kmod2 i {0, 1} is the remainder upon division of k i by 2 for i = 1, 2,...,n. Claim 2. In ZF+ AC ℵ0 +LM the chromatic number of the graph G cannot be equal to any positive integer n nor even to ℵ 0. Proof. Closely repeats the one presented for G in Section Remarks 1. It is certainly possible to construct other examples of distance graphs on R n whose chromatic number in ZFC is between 2 and 2 n, and is not defined or uncountable in ZF + AC ℵ0 + LM.

5 Note / Journal of Combinatorial Theory, Series A 110 (2005) These examples illuminate the influence of the system of axioms for set theory on combinatorial results. They also suggest that likewise the chromatic number of R n may not exist in the absolute but depend upon the system of axioms we choose for set theory. 3. See [Soi3] in 2006 for many more resultsand the history related to thisproblem and the early Ramsey Theory. 6. Open problem This series of three papers naturally raises the following open problem: Open Problem AC: For which values of n is the chromatic number χ(r n ) of the n-space R n defined in the absolute, i.e., in ZF regardless of the addition of the axiom of choice? My Conjecture is: only for n =1. Acknowledgments I thank Professor Victor Klee for his valuable suggestions for improving the presentation of this work. I am grateful to my Princeton-Math colleagues and friends for maintaining a unique creative atmosphere in the historic Fine Hall, and Fred Roberts for the tranquility of his DIMACS. As if in farewell, this work was submitted (electronically) on my last night in Princeton, from the 9th to the 10th of August, References [Chu] A. Church, P.J. Cohen and the Continuum Problem, Proceedings of International Congress of Mathematicians (Moscow-1966), Mir, Moscow, 1968, pp [Cou] D.A. Coulson, A 15-coloring of 3-space omitting distance one, Discrete Math. 256 (2002) [CFG] H.T. Croft, K.J. Falconer, R.K. Guy, Unsolved Problems in Geometry, Springer, New York, [EB] P. Erdös, N.G. de Bruijn, A colour problem for infinite graphs and a problem in the theory of relations, Indag. Math. 13 (1951) [KW] V. Klee, S. Wagon, Old and new Unsolved Problems in Plane Geometry and Number Theory, Mathematical Association of America, Washington DC, [Nec] O. Nechushtan, On the space chromatic number, Discrete Math. 256 (2002) [SS1] S. Shelah, A. Soifer, Axiom of choice and chromatic number of the plane, J. Combin. Theory Ser. A 103 (2003) [SS2] A. Soifer, S. Shelah, Axiom of choice and chromatic number: examples on the plane, J. Combin. Theory Ser. A 105 (2004) [Soi1] A. Soifer, Chromatic number of the plane & its relatives. part I: the problem & its history, Geombinatorics XII (3) (2003) [Soi2] A. Soifer, Chromatic number of the plane & its relatives, Part II: polychromatic number & 6-coloring, Geombinatorics XII (4) (2003) [Soi3] A. Soifer, Mathematical Coloring Book: Mathematics of Coloring and the Colorful Life of Its Creators, Springer, Berlin, 2006, to appear. [Sol] R.M. Solovay, A model of set theory in which every set of reals is Lebesgue measurable, Ann. Math. Ser (1970) 1 56.