1 List Coloring and Euclidean Ramsey Theory TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: A A A A Noga Alon, Tel Aviv U. Bertinoro, May 2011

2 Euclidean Ramsey Theory The Hadwiger-Nelson Problem: what is the minimum number of colors required to color the points of the Euclidean plane with no two points of distance 1 with the same color ? Equivalently: what is the chromatic number of the unit distance graph in the plane ?

3 Nelson (1950): at least 4 Isbell (1950): at most 7 Both bounds have also been proved by Hadwiger (1945)

4 By the Erdős de-Bruijn Theorem it suffices to consider finite subgraphs of the unit distance graph The lower bound (4) and the upper bound (7) have not been improved since 1945 Wormald, O’donnell: The unit distance graph contains subgraphs of arbitrarily high girth and chromatic number 4 The higher dimensional analogs have been considered as well (Frankl and Wilson) Surveys: Chilakamarri, Soifer

5 Erdős, Graham, Montgomery, Rothschild, Spencer and Straus (73,75,75): For a finite set K in the plane, let H K be the hypergraph whose set of vertices is R 2, where a set of |K| points forms an edge iff it is an isometric copy of K. Problem: What’s the chromatic number x(H K ) of H K ? (The case |K|=2 is the Hadwiger-Nelson problem)

6 Fact (EGMRSS): If K is the set of 3 vertices of an equilateral triangle, then x(H K )=2

7 Conjecture 1 (EGMRSS): For any set of 3 points K X(H K ) ≤3 Fact (EGMRSS): If K is the set of 3 vertices of an equilateral triangle, then x(H K )=2 Conjecture 2 (EGMRSS): For any set of 3 points K which is not the set of vertices of an equilateral triangle, x(H K ) ≥ 3.

8 List Coloring [ Vizing (76), Erdős, Rubin and Taylor(79) ] Def: G=(V,E) - graph or hypergraph, the list chromatic number x L (G) is the smallest k so that for every assignment of lists L v for each vertex v of G, where |L v |=k for all v, there is a coloring f of V satisfying f(v) ε L v for all v, with no monochromatic edge Clearly x(G) ≥ x L (G) for all G, strict inequality is possible

9 Fact 1: Deciding if for an input graph G, x(G) ≤ 3 is NP-complete Fact 2 (ERT): Deciding if for an input graph G, x L (G) ≤ 3 is Π 2 -complete Question 1: x L (Unit Distance Graph)=? Question 2: For a given finite K in the plane, x L (H K )=?

10 New [ A+Kostochka (11) ]: For any finite K in the plane x L (H K ) is infinite ! That is: for any finite K in the plane and for any positive integer s, there is an assignment of a list of s colors to any point of the plane, such that in any coloring of the plane that assigns to each point a color from its list, there is a monochromatic isometric copy of K

11 The reason is combinatorial: Thm 1 (A-00): For any positive integer s there is a finite d=d(s) such that for any (simple, finite) graph G with average degree at least d, x L (G)>s. Thm 2 (A+Kostochka): For any positive integers r,s there is a finite d=d(r,s) such that for any simple (finite) r-uniform hypergraph H with average (vertex)- degree at least d, x L (H)>s.

12 A hypergraph is simple if it contains no two edges sharing more than one common vertex. E.g., for r=4, the following is not allowed

13 Note: this provides a linear time algorithm to distinguish between a simple (hyper-)graph with list chromatic number at most s, and one with list chromatic number at least b(s). E.g., distinguishing between a graph G with x L (G)≤3 and one with x L (G)≥1000 is easy. There is no such known result for usual chromatic chromatic number, and it is unlikely that such a result holds (Dinur, Mosell, Regev)

14 Deriving the geometric result from the combinatorial one: Given a finite K in the plane, prove, by induction on d, that there exists a finite, simple d-regular hypergraph H d whose vertices are points in the plane in which every edge is an isometric copy of K. H d is obtained by taking |K| copies of H d-1, obtained according to a random rotation of K.

15 Example: |K|=3, H d is obtained from 3 copies of H d-1 H d-1

16 The proofs of the combinatorial results are probabilistic. Theorem 1 (graphs with large average degree have high list chromatic number) is proved by assigning to each vertex, randomly and independently, a uniform random s-subset of the set [2s]={1,2,…,2s}. It can be shown that with high probability there is no proper coloring using the lists, provided the average degree is sufficiently large.

17 Note: if the graph is a complete bipartite graph with d vertices in each vertex class, where d is arbitrarily large, there are at least s 2d potential proper colorings (with colors in {1,2,..,s} to the first vertex class, and colors in {s+1,s+2,…,2s} to the second). Each such potential coloring will come from the lists with probability 1/2 2d, hence the expected number of colorings from the lists is at least (s/2) 2d which is far bigger than 1 ! This means that a naïve computation does not suffice.

18 The idea is to first expose the lists for a randomly chosen set of a 1/d 1/2 - fraction of the vertices, and then show that if d is sufficiently large then with high probability, no coloring from these lists can be extended to a proper coloring of the whole graph using the lists of the other vertices. More precisely: G contains a subgraph H with minimum degree at least d/2. In this subgraph, let W be a random set containing a 1/d 1/2 fraction of the vertices.

19 Expose the random lists of the vertices in W. If d>20 s, say, then with high probability, at least half of the vertices of H do not belong to W and for each s-subset S of [2s], have a neighbor in W whose list is S. Let n be the number of vertices of H, and let A be a set of half of them as above. For each fixed coloring f of the vertices of W from their lists, and for each vertex v in A, there are at least s+1 colors used by f for the neighbors of v. Thus, if the list of v is contained in these s+1 colors, there will not be any proper extension of f to v.

20 It follows that for each fixed coloring f of W from the lists, the probability that W can be extended to a proper coloring of all vertices in A using their lists is at most [(1-(s+1)/4 s ] n/2. Therefore, the probability that there exists an f that can be extended to a proper coloring of H from the lists is at most 3 |W| [(1-(s+1)/4 s ] n/2 < exp (n/d 1/2 log 3-(n/2)(s+1) /4 s ) <1. ■

21 The proof of Theorem 2 (simple r-uniform hypergraphs with high average degree have high list-chromatic number) proceeds by induction on r. The induction hypothesis has to be strengthened: it is shown that if the average degree is high, then there is an assignment of s-lists so that in any coloring using the lists, a constant fraction of all edges is monochromatic.

22 This requires a decomposition result: 99% of the edges of any r-uniform hypergraph with large average degree can be decomposed into edge-disjoint subhypergraphs, each having large minimum degree which is at least 1/r times its average degree. The probabilistic estimates are strong enough to apply simultaneously to all subhypergraphs, using the same random choice.

23 Open Problems The Hadwiger-Nelson Problem: x(Unit distance graph in the plane)= ? EGMRSS: Is it true that for any non-equilateral triangle K, x(H K )=3 ? Ronsenfeld: Is x(Odd distance graph in the plane) finite ?

24 What is the smallest possible estimate for d=d(r,s) in Theorem 2 ? (simple r-uniform hypergraphs with average degree at least d have list chromatic number bigger than s). Is it r Θ(s) ? Is there an efficient (deterministic) algorithm that finds, for a given input simple (hyper-)graph with sufficiently large minimum degree, lists L v, each of size s, for the vertices, so that there is no proper coloring using the lists ? Can one give such lists and a witness showing there is no proper coloring using them ?

25 Is probability essential in proofs of this type ?