Euclidean Ramsey Theory and List Coloring

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

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

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

4. By the Erdős de-Bruijn Theorem it suffices toconsider 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 andchromatic 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 HK be thehypergraph whose set of vertices is R2, where a set of |K| points forms an edge iff it is an isometric copy of K. Problem: What’s the chromatic number x(HK) of HK ? (The case |K|=2 is the Hadwiger-Nelson problem)

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

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

8. List Coloring [ Vizing (76), Erdős, Rubin and Taylor(79) ] Def: G=(V,E) - graph or hypergraph, the listchromatic number xL(G) is the smallest kso that for every assignment of lists Lv foreach vertex v of G, where |Lv|=k for all v,there is a coloring f of V satisfying f(v) ε Lvfor all v, with no monochromatic edge Clearly x(G) ≥ xL(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,xL(G) ≤ 3 is Π2-complete Question 1: xL(Unit Distance Graph)=? Question 2: For a given finite K in the plane, xL(HK)=?

10. New [ A+Kostochka (11) ]: For any finite K in the plane xL(HK) is infinite ! That is: for any finite K in the plane and for anypositive integer s, there is an assignment of a listof 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 monochromaticisometric copy of K

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

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

13. Note: this provides a linear time algorithm todistinguish between a simple (hyper-)graphwith list chromatic number at most s, and onewith list chromatic number at least b(s). E.g., distinguishing between a graph G with xL(G)≤3 and one with xL(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 inductionon d, that there exists a finite, simple d-regular hypergraph Hd whose vertices are points in the plane in which every edge is an isometric copy of K. Hd is obtained by taking |K| copies of Hd-1,obtained according to a random rotation of K.

15. Hd-1 Example: |K|=3, Hdis obtained from 3copies of Hd-1

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

17. Note: if the graph is a complete bipartite graphwith d vertices in each vertex class, where d isarbitrarily large, there are at least s2dpotentialproper colorings (with colors in {1,2,..,s} to thefirst vertex class, and colors in {s+1,s+2,…,2s} tothe second). Each such potential coloring willcome from the lists with probability 1/22d, hencethe expected number of colorings from the listsis 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 randomlychosen set of a 1/d1/2 - fraction of the vertices, andthen 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 atleast d/2. In this subgraph, let W be a random set containinga 1/d1/2 fraction of the vertices.

19. Expose the random lists of the vertices in W. Ifd>20s, 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 fromtheir lists, and for each vertex v in A, there are atleast 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 fromthe lists, the probability that W can be extendedto a proper coloring of all vertices in A using theirlists is at most [(1-(s+1)/4s ]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)/4s]n/2 < exp (n/d1/2 log 3-(n/2)(s+1) /4s) <1. ■

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

22. This requires a decomposition result: 99% of theedges 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 toapply simultaneously to all subhypergraphs, usingthe 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-equilateraltriangle K, x(HK)=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 rΘ(s) ? Is there an efficient (deterministic) algorithmthat finds, for a given input simple (hyper-)graphwith sufficiently large minimum degree, lists Lv,each of size s, for the vertices, so that there is noproper coloring using the lists ? Can one give such lists and a witness showingthere is no proper coloring using them ?

25. Is probability essential in proofs of this type ?