aa r X i v : . [ m a t h . C O ] N ov SIZE OF COMPONENTS OF A CUBE COLORING
MARSEL MATDINOV
Abstract.
Suppose a d -dimensional lattice cube of size n d is colored in several colorsso that no face of its triangulation (subdivision of the standard partition into n d smallcubes) is colored in m + 2 colors. Then one color is used at least f ( d, m ) n d − m times. Introduction
A theorem attributed to Lebesgue asserts that if a lattice cube of dimension d is coloredin d colors then one of the colors has a connected component spanning two opposite facetsof the cube. By the standard reasoning with nerves of coverings this means that thecovering dimension of the d -dimensional cube is at least d .There arises the following natural question: What happens if the number of colors isless than d ? In [4, 1] it was conjectured that the size of a monochromatic connectedcomponent has a lower bound of order n d − m when m + 1 colors are used. For m = d − m = 1 this conjecture is proved in [4].Here we prove this conjecture in a slightly stronger form: Theorem 1.1.
Let a d -dimensional cube Q be partitioned into n d small cubes in thestandard way and then the ( m + 1) -dimensional skeleton Q m of this partition is subdividedto the triangulation T . Suppose the vertices of T (equivalently, vertices of Q m ) are coloredin several colors so that no ( m + 1) -face σ ∈ T is colored in m + 2 different colors. Thenone of the colors is used on at least f ( d, m ) n d − m vertices of T .Remark . It is also sufficient to assume that every cubical face of Q m of dimension m + 1 has at most m + 1 colors on its vertices. Such a point of view allows not to use anytriangulation T in the statement of the theorem. Remark . If a color c has several connected components then every component can beassumed to be a separate color. So we obtain a monochromatic connected set of at least f ( d, m ) n d − m vertices of T . By adjacent vertices we mean two vertices in a single ( m + 1)-face of T . This remark also remains valid if we define adjacent vertices as contained in asingle cubical ( m + 1)-face of Q m . Remark . We do not establish any explicit values for f ( d, m ). The reader may consultthe paper [3] for this information.A similar theorem was proved independently in [3]. In Section 3 we give some corollariesof Theorem 1.1 about coverings of a cube or a torus. Acknowledgments.
The author thanks Alexey Kanel-Belov for drawing attentionto this problem and numerous discussions; and thanks Roman Karasev for his help inwriting the text and translating it to English.
Mathematics Subject Classification. Proof of Theorem 1.1
Let us introduce some notation. Let C = { c , . . . , c k } be an ordered set of k + 1 colors.For every oriented k -face of T we assign +1 is its vertices are colored in the colors of C in accordance with the orientation, − C withopposite orientation, and 0 if the face is colored in other colors or some of the colors is usedmore than once. Thus we define a cochain χ ( C ) ∈ C k ( T ) and observe the coboundaryformula:(2.1) δχ ( C ) = X c k +1 χ Cc k +1 , where we sum over all the colors and by Cc k +1 we mean the concatenation of C and thenew color c k +1 . If c k +1 coincides with a color in C then χ Cc k +1 is assumed to be zero.In the rest of the proof we consider ( k + 1)-dimensional cubical subcomplexes Q k ⊆ Q m ,which are ( k + 1)-dimensional cubical skeleta of some ( d − m + k )-dimensional faces (bigfaces, not faces of a partition) of the cube Q , for k = m, m − , . . . , L : C k ( Q m ) → C k ( T ) that assigns to any k -face τ ∈ Q m the sum of simplicial k -faces of T that partition τ with appropriate orientations. Obviously L commutes with the boundary map ∂ .We are going to balance the complexes Q k as follows: Definition 2.1.
For every k -face σ ∈ Q k we will assign a ( k + 1)-chain B ( σ ) ∈ C k +1 ( T )so that for any ( k + 1)-face τ ∈ Q k and a set C of k + 1 colors the following holds:(2.2) χ C ( L ( ∂τ ) + ∂B ( ∂τ )) = 0 , where we assume that B is linearly extended to k -chains of Q k .We also put A ( σ ) = L ( σ ) + ∂B ( σ ) and use (2.2) in the following form:(2.3) χ C ( A ( ∂τ )) = 0 . Let us check that the ( m + 1)-skeleton Q m of Q is already balanced (so that we mayput B ( · ) = 0). Let | C | = m + 1, then χ C ( L ( ∂τ )) = ( δχ C , L ( τ )) = 0 , since δχ C = 0 by the formula (2.1) and the assumption of the theorem (no face of T iscolored in m + 2 different colors).The plan of the remaining part of the proof is following: • Denote the k -skeleton of any facet of the cube corresponding to Q k by Q k − ; • Balance Q k − by defining a suitable B : C k − ( Q k − ) → C k ( T ); • Make sure that in all the expressions B ( σ ) for all ( k − σ ∈ Q k − every k -face β ∈ T is used at most C ( d, k −
1) times (counted with its multiplicity inthe chains B ( σ )).If this plan passes then on the last stage we have a 1-dimensional skeleton of a ( d − m )-dimensional cube (containing n d − m small cubes) Q . To every vertex v ∈ Q we assign achain of 1-faces of T denoted by B ( v ). Then 0-chains A ( v ) are simply sets of vertices of T with integer multiplicities such that the sum of coefficients in every A ( v ) is 1. For anycolor c by (2.3) we obtain that χ c ( A ( v )) = χ c ( A ( v )) for any pair of adjacent vertices v and v . Since Q is a connected graph we obtain that the number χ c ( A ( v )) = x c does notdepend on v . The sum over all colors is X χ c ( A ( v )) = (1 , A ( v )) = 1 , IZE OF COMPONENTS OF A CUBE COLORING 3 so there exists a color c with nonzero x c . Hence this color is used in every support ofthe 0-cycle A ( v ) for every v . We have at least n d − m different choices of v and any pointcolored in c is counted at most C ′ C ( d,
0) times (here C ′ is the maximal number of 1-facesincident to a vertex in T ).So it remains to pass from the balancing of Q k to the balancing of Q k − . Note that forevery k -face τ ∈ Q k − we have to satisfy the equality (since ∂ = 0 and ∂A ( τ ) = ∂L ( τ ) = L ( ∂τ )):(2.4) χ C ( ∂A ( τ ) + ∂B ( ∂τ )) = χ C ( L ( ∂τ ) + ∂B ( ∂τ )) = 0 . In this formula A ( τ ) is already defined, and B is to be defined on ( k − Q k − .The equality (2.4) follows from the equality:(2.5) χ D ( A ( τ ) + B ( ∂τ )) = 0for every k -face τ ∈ Q k − and every set D of k + 1 colors. Indeed, using (2.1) from (2.5)we obtain:(2.6) χ C ( ∂A ( τ ) + ∂B ( ∂τ )) = ( δχ C , A ( τ ) + B ( ∂τ )) = = ( X c k χ Cc k , A ( τ ) + B ( ∂τ )) = 0 . Now we fix a set D of k + 1 colors. Define by(2.7) ξ D ( τ ) = χ D ( A ( τ ))a k -cocycle on Q k since for every ( k + 1)-face ρ ∈ Q k we have:(2.8) ξ D ( ∂ρ ) = χ D ( L ( ∂ρ ) + ∂B∂ρ ) = 0because Q k is balanced. To make this cocycle zero (as required in (2.5)) we have to assignto some ( k − σ ∈ Q k − as B ( σ ) some sets (with coefficients) of k -faces τ ∈ T .Obviously, it suffices to use only those k -faces τ ∈ T that are colored exactly in the colorsof D .The map σ ξ D ( B ( σ )) is going to be a ( k − C k − ( Q k − )with coboundary ξ D | Q k − . In order to use any k -face (out of those colored in D ) at most C ( d, k −
1) times we have to check that the ratio between the norm (sum of absolute values)of some ( k − η ∈ C k − ( Q k − ) such that δη = ξ D on Q k − and the number of k -faces usable in B ( σ ) (that is, colored in D ) is bounded by a constant C ( d, k − ξ D as an element of C k ( Q k ) equal M . By the assumption that in every B ( τ ) a ( k + 1)-face of T is used at most C ( d, k ) times we conclude that every k -face of T is used in all A ( τ ) at most C ( d, k ) C ′ ( k ) times, where C ′ ( k ) is the maximal numberof ( k + 1)-faces containing a given k -face of T (it can be bounded independently on thechoice of a particular triangulation T ). By the formula ξ D ( τ ) = χ D ( A ( τ )) we concludethat among the k -faces of T there do exist at least MC ( d,k ) C ′ ( k ) “candidates” for B ( σ ). Nowit suffices to solve the equation δη = ξ D on cochains on Q k − so that the norm | η | is atmost C ′′ ( d, k − M . After that we can assign to cubical faces of Q k − on which η isnonzero several faces of T on which ξ D is nonzero.Note that | ξ D | = M and for some codimension 1 cubical section Q ′ of Q k (parallel to Q k − ) we have: | ξ D | Q ′ | ≤ M/n . Then we use the “filling inequality” (see for example [2],where filling inequalities are widely used):
Lemma 2.2.
For a k -dimensional cocycle α on the cubical partition of the d ′ -dimensionalcube Q ′ (in terms of this proof ) there exists a ( k − -dimensional cubical cochain β suchthat δβ = α and | β | ≤ C F ( d ′ , k ) n | α | . IZE OF COMPONENTS OF A CUBE COLORING 4
By this lemma we select a ( k − β on Q ′ with norm at most M C F ( d − m + k, k ) with coboundary ξ D | Q ′ . Denote the part of ξ D between Q ′ and Q k − by ξ ′ D ; this is a cochain with norm at most M . As the required ( k − η on Q k − we may take:(2.9) η = β + π ∗ ( ξ ′ D )with norm at most ( C F ( d − m + k, k ) + 1) M . Here β is moved from Q ′ to Q k − by thetranslation and by π ∗ ( ξ ′ D ) we mean the direct image under the projection onto Q k − thatdrops the dimension by 1. The cochain π ∗ ( ξ ′ D ) can be defined explicitly (thanks to thecubical complexes that we use) as taking any ( k − σ ∈ Q k − to the sum of values(with appropriate signs) of ξ ′ D on k -faces of Q k that project onto σ . In other words: π ∗ ( ξ ′ D )( τ ) = ξ ′ D ( π − ( τ )) . So we satisfy the equality (2.5) for a particular color set D . Now we can add the chains B ( σ ) corresponding to different sets D . The expressions B ( σ ) for every particular D contained exclusively k -faces colored in the colors in D . Hence for different D we usedifferent faces, which guarantees the bounded multiplicity of the union of all B ( σ ). Thefaces of B ( σ ) colored in D do not affect the equality (2.5) for another set D ′ (not obtainedfrom D by a permutation). Now to complete the proof it remains to prove the lemma. Proof of Lemma 2.2.
The proof is similar to the proof of [3, Lemma 2.6], which is statedin terms of cycles Poincar´e dual to the cocyles in this proof.Put α = α . We are going to build a cocycle α i +1 out of α i as follows. Take a hyperplanesection Z of Q ′ parallel to a pair of its opposite facets so that | α i | Z | ≤ | α i | /n . This ispossible by the Dirichlet principle.Define a ( k − β i as follows: β i ( τ ) = α i ([ τ, π Z ( τ )]) , where [ τ, π Z ( τ )] is (at most k )-dimensional parallelepiped between τ and its projectiononto Z with appropriate sign.Now we put α i +1 = α i − δβ i and note that | β i | is at most n | α i | . Note also that α i +1 takes the same values as thetranslation of α i | Z on sections parallel to Z and is zero on any face orthogonal to Z .After several such operations for different directions of Z ( d ′ − k + 1 will be enough)the cocycle α i +1 becomes zero. We have the inequality: | α i +1 | ≤ | α i | ≤ · · · ≤ | α | . If we take β = P i β i then the required inequality holds with constant C F ( d ′ , k ) ≤ d ′ − k + 1. (cid:3) Some corollaries
We give some topological corollaries of Theorem 1.1:
Corollary 3.1. Let a d -dimensional cube Q be covered by closed sets C i so that no pointis covered more than m + 1 times. Then one of the sets C i intersects at least d − m pairsof opposite facets of Q .Remark . This is a generalization of the Lebesgue theorem. The statement of this corollary is suggested by R. Karasev as a simpler version of Corollary 3.3.
IZE OF COMPONENTS OF A CUBE COLORING 5
Proof.
We pass in a standard way from the covering to coloring the vertices of the partitionof Q into n d cubes. If the partition is fine enough then no partition face has m + 2 distinctcolors.Let us include Q into the cube 2 Q of size (2 n ) d and repeat the coloring of Q usingreflections with respect to the halving hyperplanes of 2 Q . Then we extend the coloringonto the whole Z d with translations by ± n along the coordinate axes. Let us see whathappens with a color c i . Following Remark 1.3 we assume that the color c i makes aconnected subset of Q . The vertices of Z d colored in c i can be decomposed into connectedcomponents; denote one of them by c ′ i . If the component c i spans a pair of opposite facets(orthogonal to a base vector e j ) in Q then c ′ i is invariant under the translation by ± ne j .Otherwise c i does not touch one of the facets orthogonal to e j and c ′ i is trapped betweena pair of hyperplanes orthogonal to e j at distance 2 n from each other.So the free Abelian group Λ i of translational symmetries of c ′ i has dimension exactly ℓ , where ℓ is the number of pairs of opposite facets of Q intersected by c i and c ′ i can beobtained from 2 Q ∩ c ′ i by translations in Λ i . If we intersect c ′ i with a large cube Q ′ ofsize (2 nN ) d then the cardinality of c ′ i ∩ Q ′ has the growth order N ℓ for varying N . ByTheorem 1.1 and Remark 1.3 some c ′ i ∩ Q ′ must have the number of vertices of order atleast N d − m ; so for some of c ′ i we must have ℓ ≥ d − m . (cid:3) Corollary 3.3.
Let a d -dimensional torus T d be covered by open sets C i so that no pointis covered more than m + 1 times. Then for some C i the image of H ( C i ) in H ( T d ) = Z d has dimension at least d − m .Remark . The sets have to be open so that the connectedness and the arcwise con-nectedness coincide.
Proof.
As in the previous proof we pass from the covering of T d to a fine enough triangu-lation of a covering cube Q , which subdivides the cubical partition into n d small cubes.Then we assume that the vertices of the triangulation are colored so that no face hasmore than m + 1 colors. Duplicating Q by translations we obtain a large cube Q N withside length N n and the corresponding coloring. By gluing the opposite facets of Q N weobtain a torus naturally N d -fold covering T d .By Theorem 1.1 in Q N we have a monochromatic connected component S with sizeof order N d − m . Let S be the maximal intersection of S with a residue class modulo Q (points are equal modulo Q if the differences of their coordinates are divisible by n ). Then | S | ≥ | S | /n d ≥ N d − m n d .Note that a projection of a monochromatic path in S starting in a point of S withcoordinates ( x , . . . , x d ) and ending in a point of S with coordinates ( y , . . . , y d ) is amonochromatic closed loop in T d representing the homology class ( y i − x i n ) i . So it sufficesto show that the dimension of the linear space generated by pairwise differences of S is atleast d − m (the covering set corresponding to S will be the one required). Equivalently,we have to show that the dimension of the affine hull of S is at least d − m .Assume the contrary: The dimension of the affine hull of S is at most d − m −
1. Then S is contained in at most ( d − m − L and the number ofvertices in Q N ∩ L (and therefore the number of vertices in S ) is at most ( N n ) d − m − .For large enough N we obtain a contradiction with the inequality | S | ≥ N d − m n d . (cid:3) References [1] A. Belov-Kanel, I. Ivanov-Pogodaev, A. Malistov, M. Kharitonov. Colorings and clusters. // 22summer conference International mathematical Tournament of towns, Teberda, Karachai-Cherkess,02.08.2010–10.08.2010, olympiads.mccme.ru/lktg/2010/2/2-1en.pdf, 2010.
IZE OF COMPONENTS OF A CUBE COLORING 6 [2] M. Gromov. Singularities, expanders, and topology of maps. Part 2: from combinatorics to topologyvia algebraic isoperimetry. // Geometric and Functional Analysis 20:2 (2010), 416–526.[3] R. Karasev. An analogue of Gromov’s waist theorem for coloring the cube. // arXiv:1109.1078,(2011).[4] J. Matouˇsek and A. Pˇr´ıv˘etiv´y. Large monochromatic components in two-colored grids. // SIAM J.Discrete Math. 22:1 (2008), 295–311.
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