The measurable chromatic number of Euclidean space Christine Bachoc - PowerPoint PPT Presentation

The measurable chromatic number of Euclidean space Christine Bachoc Universit´ e Bordeaux I, IMB Codes, lattices and modular forms Aachen, September 26-29, 2011

χ ( R n ) ◮ The chromatic number of Euclidean space χ ( R n ) is the smallest number of colors needed to color every point of R n , such that two points at distance apart 1 receive different colors. ◮ E. Nelson, 1950, introduced χ ( R 2 ) . ◮ Dimension 1: χ ( R ) = 2 ◮ No other value is known!

χ ( R 2 ) ≤ 7

χ ( R 2 ) ≤ 7

χ ( R 2 ) ≤ 7

χ ( R 2 ) ≥ 4 Figure: The Moser’s Spindle

The two inequalities: 4 ≤ χ ( R 2 ) ≤ 7 where proved by Nelson and Isbell, 1950. No improvements since then...

χ ( R n ) ◮ Other dimensions: lower bounds based on χ ( R n ) ≥ χ ( G ) for all finite graph G = ( V , E ) embedded in R n ( G ֒ → R n ) i.e. such that V ⊂ R n and E = { ( x , y ) ∈ V 2 : � x − y � = 1 } . ◮ De Bruijn and Erd¨ os (1951): χ ( R n ) = max χ ( G ) G finite → R n G ֒ ◮ Good sequences of graphs: Raiski (1970), Larman and Rogers (1972), Frankl and Wilson (1981), Sz´ ekely and Wormald (1989).

χ ( R n ) for large n ( 1 . 2 + o ( 1 )) n ≤ χ ( R n ) ≤ ( 3 + o ( 1 )) n ◮ Lower bound : Frankl and Wilson (1981). Use graphs with vertices in { 0 , 1 } n and the “linear algebra method” to estimate χ ( G ) . ◮ FW 1 . 207 n is improved to 1 . 239 n by Raigorodskii (2000). ◮ Upper bound: Larman and Rogers (1972). Use Vorono¨ ı decomposition of lattice packings.

χ m ( R n ) ◮ The measurable chromatic number χ m ( R n ) : the color classes are required to be measurable. ◮ Obviously χ m ( R n ) ≥ χ ( R n ) . ◮ Falconer (1981): χ m ( R n ) ≥ n + 3. In particular χ m ( R 2 ) ≥ 5 ◮ The color classes are measurable 1-avoiding sets, i.e. contain no pair of points at distance apart 1.

m 1 ( R n ) � � m 1 ( R n ) = sup δ ( S ) : S ⊂ R n , S measurable, avoids 1 where δ ( S ) is the density of S : vol ( S ∩ B n ( r )) δ ( S ) = lim sup . vol ( B n ( r )) r → + ∞ δ = 1 / 7

m 1 ( R n ) ◮ Obviously 1 χ m ( R n ) ≥ m 1 ( R n ) ◮ Problem: to upper bound m 1 ( R n ) . ◮ Larman and Rogers (1972): m 1 ( R n ) ≤ α ( G ) → R n for all G ֒ | V | where α ( G ) is the independence number of the graph G i.e. the max number of vertices pairwise not connected by an edge.

Finite graphs ◮ An independence set of a graph G = ( V , E ) is a set of vertices pairwise not connected by an edge. ◮ The independence number α ( G ) of the graph is the number of elements of a maximal independent set. ◮ A 1-avoiding set in R n is an independent set of the unit distance graph V = R n E = { ( x , y ) : � x − y � = 1 } .

1-avoiding sets versus packings S avoids d = 1 δ ( S ) = lim vol ( S ∩ B n ( r )) vol ( B n ( r )) m 1 ( R n ) = sup S δ ( S ) ? S avoids d ∈ ] 0 , 2 [ δ ( S ) = lim | S ∩ B n ( r ) | vol ( B n ( r )) δ n = sup S δ ( S ) ? S avoids d = 1 δ ( S ) = | S | | V | α ( G ) | V | = sup S δ ( S ) ?

The linear programming method ◮ A general method to obtain upper bounds for densities of distances avoiding sets. ◮ For packing problems: initiated by Delsarte, Goethels, Seidel on S n − 1 (1977); Kabatianskii and Levenshtein on compact 2-point homogeneous spaces (1978); Cohn and Elkies on R n (2003). ◮ For finite graphs: Lov´ asz theta number ϑ ( G ) (1979). ◮ For sets avoiding one distance: B, G. Nebe, F . Oliveira, F . Vallentin for m ( S n − 1 , θ ) (2009). F . Oliveira and F . Vallentin for m 1 ( R n ) (2010).

Lov´ asz theta number ◮ The theta number ϑ ( G ) (L. Lov´ asz, 1979) satisfies the Sandwich Theorem: α ( G ) ≤ ϑ ( G ) ≤ χ ( G ) ◮ It is the optimal value of a semidefinite program ◮ Idea: if S is an independence set of G , consider the matrix B S ( x , y ) := 1 S ( x ) 1 S ( y ) / | S | . B S � 0, B S ( x , y ) = 0 if xy ∈ E , | S | = � ( x , y ) ∈ V 2 B S ( x , y ) .

ϑ ( G ) ◮ Defined by: � � : B ∈ R V × V , B � 0 , ϑ ( G ) = max B ( x , y ) ( x , y ) ∈ V 2 � B ( x , x ) = 1 , x ∈ V � B ( x , y ) = 0 xy ∈ E ◮ Proof of α ( G ) ≤ ϑ ( G ) : Let S be an independent set. B S ( x , y ) = 1 S ( x ) 1 S ( y ) / | S | satisfies the constraints of the above SDP . Thus � B S ( x , y ) = | S | ≤ ϑ ( G ) . ( x , y ) ∈ V 2

ϑ ( R n ) ◮ Over R n : take B ( x , y ) continuous, positive definite, i.e. for all k , for all x 1 , . . . , x k ∈ R n , � � B ( x i , x j ) 1 ≤ i , j ≤ k � 0. ◮ Assume B is translation invariant: B ( x , y ) = f ( x − y ) (the graph itself is invariant by translation). ◮ Replace � ( x , y ) ∈ V 2 B ( x , y ) by 1 � δ ( f ) := lim sup f ( z ) dz . vol ( B n ( r )) r → + ∞ B n ( r )

ϑ ( R n ) ◮ Leads to: � ϑ ( R n ) f ∈ C b ( R n ) , f � 0 := sup δ ( f ) : f ( 0 ) = 1 , � f ( x ) = 0 � x � = 1 Theorem (Oliveira Vallentin 2010) m 1 ( R n ) ≤ ϑ ( R n )

The computation of ϑ ( R n ) ◮ Bochner characterization of positive definite functions: � f ∈ C ( R n ) , f � 0 ⇐ R n e ix · y d µ ( y ) , µ ≥ 0 . ⇒ f ( x ) = ◮ f can be assumed to be radial i.e. invariant under O ( R n ) : � + ∞ f ( x ) = Ω n ( t � x � ) d α ( t ) , α ≥ 0 . 0 where Ω n ( t ) = Γ( n / 2 )( 2 / t ) ( n / 2 − 1 ) J n / 2 − 1 ( t ) . ◮ Then take the dual program.

The computation of ϑ ( R n ) ◮ Leads to: ϑ ( R n ) � = inf z 0 : z 0 + z 1 ≥ 1 z 0 + z 1 Ω n ( t ) ≥ 0 for all t > 0 } ◮ Explicitly solvable. For n = 4, graphs of Ω 4 ( t ) and of the optimal function f ∗ 4 ( t ) = z ∗ 0 + z ∗ 1 Ω 4 ( t ) : The minimum of Ω n ( t ) is reached at j n / 2 , 1 the first zero of J n / 2 .

The computation of ϑ ( R n ) ◮ We obtain n ( t ) = Ω n ( t ) − Ω n ( j n / 2 , 1 ) − Ω n ( j n / 2 , 1 ) f ∗ ϑ ( R n ) = 1 − Ω n ( j n / 2 , 1 ) . 1 − Ω n ( j n / 2 , 1 ) ◮ Resulting upper bound for m 1 ( R n ) (OV 2010): − Ω n ( j n / 2 , 1 ) m 1 ( R n ) ≤ ϑ ( R n ) = 1 − Ω n ( j n / 2 , 1 ) ◮ Decreases exponentially but not as fast as Frankl Wilson Raigorodskii bound (1 . 165 − n instead of 1 . 239 − n ). A weaker bound, but with the same asymptotic, was obtained in BNOV 2009 through m ( S n − 1 , θ ) .

ϑ G ( R n ) ◮ To summarize, we have seen two essentially different bounds: m 1 ( R n ) ≤ α ( G ) with FW graphs and lin. alg. bound | V | m 1 ( R n ) ≤ ϑ ( R n ) → R n morally encodes ϑ ( G ) for every G ֒ ◮ The former is the best asymptotic while the later improves the previous bounds in the range 3 ≤ n ≤ 24. ◮ It is possible to combine the two methods, i.e to insert the constraint relative to a finite graph G inside ϑ ( R n ) . Joint work (in progress) with F . Oliveira and F . Vallentin.

ϑ G ( R n ) → R n , for x i ∈ V , let r i := � x i � . Let G ֒ α ( G ) ϑ G ( R n ) := inf { z 0 + z 2 | V | : z 2 ≥ 0 z 0 + z 1 + z 2 ≥ 1 � | V | z 0 + z 1 Ω n ( t ) + z 2 ( 1 i = 1 Ω n ( r i t )) ≥ 0 | V | for all t > 0 } . Theorem m 1 ( R n ) ≤ ϑ G ( R n ) ≤ ϑ ( R n )

Sketch of proof ◮ ϑ G ( R n ) ≤ ϑ ( R n ) is obvious: take z 2 = 0. ◮ Sketch proof of m 1 ( R n ) ≤ ϑ G ( R n ) : let S a measurable set avoiding 1. Let f S ( x ) := δ ( 1 S − x 1 S ) . δ ( S ) f S is continuous bounded, f S � 0, f S ( 0 ) = 1, f S ( x ) = 0 if � x � = 1. Moreover δ ( f S ) = δ ( S ) . ◮ Thus f S is feasible for ϑ ( R n ) , which proves that δ ( S ) ≤ ϑ ( R n ) .

Sketch of proof ◮ If V = { x 1 , . . . , x M } , for all y ∈ R n , M � 1 S − x i ( y ) ≤ α ( G ) . i = 1 ◮ Leads to the extra condition: M � f S ( x i ) ≤ α ( G ) . i = 1 ◮ Design a linear program, apply Bochner theorem, symmetrize by O ( R n ) , take the dual.

ϑ G ( R n ) ◮ Bad knews: cannot be solved explicitly (we don’t know how to) ◮ Challenge: to compute good feasible functions. ◮ First method: to sample an interval [ 0 , M ] , solve a finite LP , then adjust the optimal solution (OV, G = simplex). Figure: f ∗ 4 ( t ) (blue) and f ∗ 4 , G ( t ) (red) for G = simplex

ϑ G ( R n ) ◮ Observation: the optimal has a zero at y > j n / 2 , 1 . ◮ Idea: to parametrize f = z 0 + z 1 Ω n ( t ) + z 2 Ω n ( rt ) with y : f ( y ) = f ′ ( y ) = 0, f ( 0 ) = 1 determines f . ◮ We solve for:  z 0 + z 1 + z 2 = 1   z 0 + z 1 Ω n ( y ) + z 2 Ω n ( ry ) = 0  z 1 Ω ′ n ( y ) + rz 2 Ω ′ n ( ry ) = 0  ◮ Then, starting with y = j n / 2 , 1 , we move y to the right until f y ( t ) := z 0 ( y ) + z 1 ( y )Ω n ( t ) + z 2 ( y )Ω n ( rt ) takes negative values.