Odd distances in colourings of the plane

We prove that every finite colouring of the plane contains a monochromatic pair of points at an odd distance from each other.


Introduction
The odd distance graph is the graph with vertex set R 2 , and whose edges are the pairs of points in R 2 whose Euclidean distance is an odd integer.The chromatic number of a graph is equal to the minimum number of colours required to assign each vertex a colour so that no two adjacent vertices receive the same colour.In 1994, Rosenfeld posed to Erdős the problem of determining the chromatic number of the odd distance graph, and Erdős [11] presented the problem the next day in an open problems talk.Since then, the problem has been reiterated numerous times (see for instance, [1-3, 5, 6, 14-20, 23-27, 29-31]).
At the time the best known lower bound was 4, the same as for the Hadwiger-Nelson problem.In 2009, Ardal, Maňuch, Rosenfeld, Shelah, and Stacho [1] constructed a 5-chromatic subgraph of the odd distance graph.Making a breakthrough in the Hadwiger-Nelson problem, de Grey [8] further constructed a 5-chromatic unit distance graph.Motivated by work towards constructing a 6-chromatic unit distance graph, Heule [16,17] recently offered $500 for determining whether or not the odd distance graph is 5-colourable.The prize was promptly claimed by Parts [23] with a construction of a 6-chromatic subgraph of the odd distance graph.
As conjectured by Soifer [30], we prove that the odd distance graph has no finite colouring.
Theorem 1.For every finite colouring of the plane, there is a monochromatic pair of points at an odd distance from each other.
Since there are countable colourings of the odd distance graph, the chromatic number of the odd distance graph is therefore countably infinite.This solves Rosenfeld's odd distance problem.An example of such a countable colouring is given by assigning the points [ 1  2 x, 1 2 (x + 1)) × [ 1 2 y, 1 2 (y + 1)) ⊂ R 2 the colour (x, y) for each (x, y) ∈ Z 2 .In fact, we prove the following more general result.
Theorem 2. For every finite colouring of the plane, for every pair of integers p, q, there is a monochromatic pair of points at a distance congruent to p (mod q) from each other.
Very recently, with Rose McCarty and Micha l Pilipczuk [7] we have extended the methods in the present paper to prove two generalizations of Theorem 1 for prime and polynomial distances.
Our construction may also be of independent interest from a purely graph theoretic point of view.Indeed, when q does not divide p, the graphs we construct to prove Theorem 2 happen to be triangle-free and appear to not contain any of the classical constructions of triangle-free graphs with large chromatic number (see [28] for a discussion of such constructions).
If we restrict our colourings of the plane to be measurable, then much more than Theorem 2 is known.Fürstenberg, Katznelson, and Weiss [13] gave an ergodic-theoretic proof that for every finite measurable colouring of the plane, there exists a d 0 > 0 such that for every real d ≥ d 0 , there is a monochromatic pair of points at a distance of d from each other.Later, Bourgain [4] gave a harmonic-analytic proof, Falconer and Marstrand [12] gave a more direct geometric proof, and Oliveira and Vallentin [9] gave a proof using Fourier analysis.
Steinhardt [31] gave a spectral proof of Theorem 1 under the additional assumption of the colouring being measurable.Inspired by this, our proof of Theorem 2 also uses spectral techniques.In Section 2 we prove an analogue of the Lovász theta bound [22] for Cayley graphs of Z d (see Theorem 3).Then in Section 3, we prove Theorem 2 by constructing Cayley graphs of Z 2 (see Theorem 8) which can both be realized as distance graphs in the plane, and for which we can effectively apply Theorem 3.

Ratio bound
We say that a set C ⊆ Z d is centrally symmetric if C = −C.Similarly, a function w : C → R is centrally symmetric if w(x) = w(−x) for all x ∈ C. For a centrally symmetric set C ⊆ Z d \{0}, we let G(Z d , C) be the Cayley graph of Z d with generating set C. This is the graph with vertex set Z d , where two vertices u, v ∈ Z d are adjacent if there exists a c ∈ C with u + c = v.Given a real symmetric matrix B, we let λ max (B) and λ min (B) be equal to its maximum and minimum eigenvalue respectively.For x, y ∈ Z d , we write x • y for the dot product.
An independent set of a graph G is a set of pairwise non-adjacent vertices.The independence number of a graph G is equal to the maximum size of an independent set in G.For a graph G, the independence number and chromatic number of G are denoted by α(G) and χ(G) respectively.For a finite graph G, we let α(G) = α(G)/|V (G)|.Notice that χ(G) ≥ 1/α(G).For Cayley graphs on Z d , we define α(G) in terms of the maximum upper density of an independent set.Given a set I ⊆ Z d , its upper density is For a Cayley graph G(Z d , C), we let The purpose of this section is to prove the following analogue of the Lovász theta bound [22].
Theorem 3. Let C ⊆ Z d \{0} and w : C → R be centrally symmetric with x∈C w(x) absolutely convergent and positive.Then , We remark that one could derive Theorem 3 from a measurable analogue for Cayley graphs G(R d , C) due to Bachoc, DeCorte, Oliveira, and Vallentin [2].Indeed, Dutour Sikirić, Madore, Moustrou, and Vallentin [10] use this to derive a slightly less general version of Theorem 3 which gives a bound for the chromatic number of the Voronoi tessellation of a lattice.For ease of accessibility, we will instead prove Theorem 3 directly from Lovász's [22] theta bound for finite graphs, and a simple lemma computing the eigenvalues of a weighted adjacency matrix of a Cayley graph G(Z d n , C), where Z d n is the abelian group (Z/nZ) d .Theorem 4 (Lovász [22]).If G is a finite graph, then , Lemma 5 (Lovász [21]).Let w : Z d n → R be centrally symmetric, and let B be the symmetric matrix in We now derive Theorem 3 from Theorem 4 and Lemma 5.
Proof of Thereom 3. Since x∈C w(x) is absolutely convergent and positive, we have x∈C |w(x)| = s for some finite real s > 0. For each real 0 < ǫ < s, let C ǫ be a finite subset of C such that x∈Cǫ |w(x)| > s − ǫ.For n sufficiently large, let w ǫ : Z d n → R be such that w ǫ (x) = 0 for x ∈ C ǫ , and w ǫ (x) = w(x) for x ∈ C ǫ , and let B ǫ,n be the symmetric matrix in By Lemma 5, the eigenvalues of B ǫ,n are {λ z : z ∈ Z d n }, where for each z ∈ Z d n , So, it follows that lim n→∞ λ max (B ǫ,n ) = sup u∈(R/Z) d w ǫ (u), and lim n→∞ λ min (B ǫ,n ) = inf u∈(R/Z) d w ǫ (u).In particular, by Theorem 4, for each real 0 < ǫ < s, we have that .
Consider some real 0 < ǫ < s.Then, since |C ǫ | is finite, there exists some positive integer T such that C ǫ ⊂ [−T, T ] d .For every real ǫ ′ > 0 and positive integer N , there exists an independent set Since ǫ ′ > 0 can be chosen arbitrarily small and lim n→∞ Finally, putting all of this together and examining limits, we get where the last line follows since for all 0 < ǫ < s and u ∈ (R/Z) d , we have that w(u) − ǫ < w ǫ (u) < w(u) + ǫ.

Distance graphs
Theorem 2 is trivial in the case that p ≡ 0 (mod q), since qZ × {0} ⊂ R 2 would be an infinite clique.So, after scaling, it is enough to prove Theorem 2 for coprime positive integers p, q with p < q.We start this section by choosing generating sets for our Cayley graphs of Z 2 .The generating sets are chosen carefully to ensure that our Cayley graphs have large chromatic number, and that they still embed as a distance graph in the plane.
For each triple of positive integers p, q, k with p, q coprime and p < q, let D p,q,k = {±(q 2k+t − q 2k−t , −pq t − 2q 2k ) : t = 0, 1, . . ., 2k − 1}, and The key property of the set D p,q,k that ensures that G(Z 2 , C p,q,k ) has large chromatic number is essentially the diversity of the highest power of q that divides the two coordinates of the different points of D p,q,k ; as t increases, the highest power of q dividing the first coordinate decreases, and the highest power of q dividing the second coordinate increases.Intuitively, this is at least enough to avoid simple periodic colourings with a small period (depending on k).We remark that if we defined D p,q,k more simply to be the set {±(q 2k−t , q t ) : t = 0, 1, . . ., 2k − 1}, then G(Z 2 , C p,q,k ) would still have large chromatic number, however, then we would be unable to show that G(Z 2 , C p,q,k ) has a suitable embedding as a distance graph in the plane.
First, let us show that G(Z 2 , C p,q,k ) has an embedding as a distance graph in the plane.
Next, we will focus on applying Theorem 3 to show that G(Z 2 , C p,q,k ) has large chromatic number for large k.For our arguments, it is helpful to consider finite subsets of C p,q,k .For each quadruple of positive integers p, q, k, n with p, q coprime and p < q, let C p,q,k,n = n−1 j=0 (jq + 1)D p,q,k , and let w p,q,k,n : C p,q,k,n → R be such that for each c ∈ C p,q,k,n of the form (jq + 1)d with d ∈ D p,q,k , we have w p,q,k,n (c) = n−j n(n+1) .As in Theorem 3, for u ∈ (R/Z) 2 , let w p,q,k,n (u) = x∈C p,q,k,n w p,q,k,n (x)e 2πiu•x .
So in order to effectively apply Theorem 3, it remains to bound inf u∈(R/Z) d w p,q,k,n (u).
Lemma 7.For all positive integers p, q, k with p, q coprime and p < q, we have Proof.Since (R/Z) 2 is compact, for each positive integer n, there exists some (a n , b n ) ∈ (R/Z) 2 such that w p,q,k,n ((a n , b n )) = inf u∈(R/Z) 2 w p,q,k,n (u).Furthermore, by the Bolzano-Weierstrass theorem, there is a strictly increasing sequence of positive integers Let, D 7.1 p,q,k = {(x, y) ∈ D p,q,k : q(ax + by) ≡ 0 (mod 1), ax + by ≡ 0 (mod 1)}, D 7.2 p,q,k = {(x, y) ∈ D p,q,k : q(ax + by) ≡ 0 (mod 1)}, D 7.3 p,q,k = {(x, y) ∈ D p,q,k : ax + by ≡ 0 (mod 1)}.Observe that We will obtain bounds for each of the summations over D 7.1 p,q,k , D 7.2 p,q,k , D 7.3 p,q,k individually.Note that each of D 7.1 p,q,k , D 7.2 p,q,k , D 7.3 p,q,k are centrally symmetric, so each of these three summations is real.We begin with bounding the summation over D 7.1 p,q,k .Our careful choice of D p,q,k is what enables us to obtain such a bound for this summation.
Next we tackle the summation over D 7.2 p,q,k .It is simply the natural choice of C p,q,k,n given D p,q,k that enables us to obtain such an estimate for this summation.
Choose a positive integer m arbitrarily.For each s ∈ {1, . . ., m}, let 1 − e 2πi|Ms|q(an f x+bn f y) 1 − e 2πiq(an By definition of D 7.2 p,q,k , we have q(ax + by) ≡ 0 (mod 1).So there exists some ǫ > 0 and F ∈ N, such that q(a n f x + b n f y) ∈ [ǫ, 1 − ǫ] (mod 1) for all f ≥ F .Since 1 − e 2πiq(an f x+bn f y) is bounded away from 0 for f ≥ F , we get that lim Since m was chosen arbitrarily, the claim now follows.
Lastly, we handle the summation over D 7.3 p,q,k .Here, the choice of our function w p,q,k,n , given C p,q,k,n is what enables the following bound.
Proof.Since D 7.3 p,q,k ⊆ D p,q,k is a finite centrally symmetric set, by the identity e iθ + e −iθ = 2 cos(θ), it is enough to show that if (x, y) ∈ D 7.3 p,q,k , then lim inf So, we may assume that θ f = 0. Observe that lim Let us evaluate the above integral, we substitute ℓ = 2πn f qθ f for convenience (note that ℓ = 0), The claim now follows since as desired.
We are now ready to prove that G(Z 2 , C p,q,k ) has large chromatic number (when k is large).Theorem 8.For all positive integers p, q, k with p, q coprime and p < q, we have α(G(Z 2 , C p,q,k ) ≤ 1 k+1 .As a consequence, χ(G(Z 2 , C p,q,k )) ≥ k + 1.
As previously discussed at the start of the section, Theorem 2 is trivial in the case that p ≡ 0 (mod q), since qZ × {0} ⊂ R 2 would be an infinite clique.By scaling, we may then reduce to the case that p and q are coprime.So, Theorem 2 now follows from Theorem 8 and Lemma 6.
We finish the paper with a proof that the graphs in Theorem 8 are triangle-free, and thus provide a new construction of triangle-free graphs with arbitrarily large chromatic number.Proposition 9.For all positive integers p, q, k with p, q coprime and p < q, the graph G(Z 2 , C p,q,k ) is triangle-free.