An Analogue of Gromov’s Waist Theorem for Coloring the Cube
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Abstract
It is proved that if we partition a ddimensional cube into \(n^d\) small cubes and color the small cubes in \(m+1\) colors then there exists a monochromatic connected component consisting of at least \(f(d, m) n^{dm}\) small cubes. Another proof of this result is given in Matdinov’s preprint (Size of components of a cube coloring, arXiv:1111.3911, 2011)
Keywords
Graph coloring Covering dimension Waist inequality1 Introduction
One possible way to express that \(Q^d = [0,1]^d\) has dimension d is to claim that it cannot be colored in d colors with arbitrarily small monochromatic connected components. In [9] the following related question was studied:
Question 1.1
If we color \(Q^d\) in \(m+1\) colors (to make the problem discrete we color small cubes of the partition of \(Q^d\) into \(n^d\) small cubes) then what size of a monochromatic connected component can we guarantee?
For \(m=d1\) the HEX lemma [2] (see also [9] for the details) shows that there must exist a monochromatic connected component spanning two opposite facets, and therefore consisting of at least n small cubes. The corresponding continuous result for a covering of the cube by d closed sets in place of a coloring in d colors is known for a long time; it is usually attributed to Lebesgue or may be deduced from the more general theorem in [11] about the Alexandrov width (waist), see also the discussion in [6, Section 6]. In [9] the coloring in 2 colors was studied using isoperimetric inequalities for the grid and a lower bound \(n^{d1}  d^2n^{d2}\) for the size of a monochromatic connected component was established. It was conjectured in [9] that the size of a monochromatic connected component is of order \(n^{dm}\) for \(m+1\) colors.
BelovKanel also posed the same problem in 1990s (private communication) and it circulated among mathematicians in Moscow, see for example [1, Problem 15]. Matdinov [8] has independently obtained another solution for this problem.
Now let us state the result:
Theorem 1.2
Remark 1.3
In this theorem two small cubes are connected if they have a nonempty intersection as closed sets. From the proof it becomes clear that the connectedness can also be understood in terms of 1skeleton of a triangulation subdividing the cubic partition, as it was in [9].
Remark 1.4
Matdinov [8] noted that the number of colors can be arbitrary, the only thing we have to check is that no point is colored in more that \(m+1\) colors. Indeed, the proof in Sect. 3 of this paper uses the multiplicity of coloring and does not use the total number of colors.
Remark 1.5
From (3.4) we see that for sufficiently large n we can take \(\big ((m+1)! \genfrac(){0.0pt}{}{d}{m} 4^m\big )^{1}\) as the coefficient in this theorem. The coefficient \((m+1)!\) is an inevitable consequence of the applied technique, as it was in [4, 7], while the part \(\genfrac(){0.0pt}{}{d}{m} 4^m\) still may be improved. It would be natural to have an estimate not depending on d, as it is for \(m=1\) and \(m=d1\).
Remark 1.6
From Lemma 3.1 below it will be clear that the cubic partition may not be the optimal one in this problem. In the proof of Theorem 1.2 the partition has to be put into a general position thus spoiling the constant \(f(d, m)\). A simple (which is Poincaré dual to simplicial) partition of \(Q^d\) into small parts of bounded complexity may be more suitable. We could also start (following [9]) from a partition Poincaré dual (which means consisting of stars of vertices in the barycentric subdivision) to a triangulation of \(Q^d\) obtained by triangulating every small cube of the standard cubic partition; such a partition is already simple.
The result of this paper and the method of its proof has very much in common with the results in [3, 4, 10] on “waists”. Let us remind two of them:
Theorem 1.7
Corollary 1.8
In particular, for generic smooth maps \(f: S^d\rightarrow \mathbb{R }^m\) it follows that for some point \(z\in \mathbb{R }^m\) the preimage \(f^{1}(z)\) has \((dm)\)dimensional volume \(\mathrm{vol}_{dm} f^{1}(z) \ge \mathrm{vol}_{dm} S^{dm}\).
Since \(Q^d\) can be embedded into \(S^d\) preserving the volumes up to some constant, Theorem 1.2 could follow from Corollary 1.8 directly if there were some lower bounds for maximal volume of a connected component of \(f^{1}(z)\). An open problem is to establish such a result.
2 Filling of Cycles in \((Q, \partial Q)\)
Fix the dimension d and denote \(Q^d\) by Q for brevity. Generally, we are going to follow the approach of [4] to examine the filling profile for cycles in \((Q,\partial Q)\). Let us work with singular kdimensional chains \(C_k(Q)\) of the space Q, represented by piecewise linear (or shortly PL) images of kdimensional simplicial complexes. Assume that Q is the unit cube in the Euclidean space.
Definition 2.1
The well known fact is that \(H_k(Q,\partial Q) = 0\) for \(k\ne d\) and \(H_d(Q,\partial Q)=\mathbb{Z }\).
Remark 2.2
In the rest of the proof we use integral homology and have to be careful about signs. Though everything passes literally with modulo 2 homology without worrying about signs; the integral homology is just left for other possible applications.
So the operator \(\partial \) is invertible on \(Z_k(Q,\partial Q)\) for \(k<d\) and the following lemma allows to invert it economically:
Lemma 2.3
Remark 2.4
The value \(H(z)\) may not depend on z linearly, but the essential thing is that we have a linear bound on the volume.
Proof
Definition 2.5
Call a chain \(c\in C_k(Q)\) rectilinear if all its simplexes are parallel to coordinate ksubspaces of \(\mathbb{R }^d\).
For this type of chains we improve the filling profile:
Lemma 2.6
Proof
3 Proof of Theorem 1.2
Denote by \(C_1, \ldots , C_M\) the monochromatic connected components of the coloring. By perturbing slightly the walls of the partition of Q into small cubes we may assume that every intersection \(C_{i_0}\cap \cdots \cap C_{i_k}\) is either empty or has dimension precisely \(dk\). In this partition of \(Q\) into parts some sets \(C_i\) may become disconnected but this makes the statement even stronger.

The coloring corresponds to a PL map \(\chi : Q\rightarrow \Delta \), where \(\Delta \) is an mdimensional simplex with vertices \(w_0,\ldots , w_m\). The subset of Q of color i corresponds to the preimage of the star of \(v_i\) in the barycentric subdivision \(\Delta ^{\prime }\). If we perturb the map \(\chi \) so that it becomes transversal to the triangulation \(\Delta ^{\prime }\) then we obtain the required properties of intersections \(C_{i_0}\cap \cdots \cap C_{i_k}\) because they correspond to preimages of codimension k faces of \(\Delta ^{\prime }\).

We can also view the partition of Q into smaller cubes as projection of facets of the graph of a convex function \({\varphi } : Q \rightarrow \mathbb{R }\) onto Q. This function can be built as follows: take the convex PL function of one variable \({\varphi }_1(x)\) with discontinuity of the derivative exactly at \(x=1/n, 2/n,\ldots , (n1)/n\). Then \({\varphi }(x_1,\ldots , x_d) = \sum _{i=1}^d {\varphi }_1(x_i)\) is exactly what is needed. Then we perturb the graph of \({\varphi }\) to put its facets into general position (thus making this graph simple) and obtain the PL function \(\psi (x)\). The projections of facets of the graph of \(\psi \) give a partition of Q arbitrarily close to the cubic partition. The advantages of this method are convexity of parts and reasonable behaviour of faces of the partition in any dimension.

The third way to make the partition simple is shifting of the partition cubes.^{1} Suppose in every layer of \(n^{d1}\) cubes we have shifted the cubes so that the corresponding \((d1)\)dimensional pattern is simple. Then we shift layers relative to each other and the general position will be simple again. Under such a shifting some cubes in the partition will be cut out and we will have to introduce some new (incomplete) partition cubes. This does not affect the proof because we may assume that the volumes of all the faces of the partition change by arbitrarily small value. Important thing is that all intersections in this partition remain rectilinear and we may apply the improved filling profile from Lemma 2.6.

Another way of making the partition simple ^{2} is to notice that the cubic partition is the Voronoi partition corresponding to the integer grid. Now we can deform the Euclidean norm (keeping the points) so that the corresponding partition becomes simple.
Assume that the volume of every \(C_i\) is at most \(\alpha n^{m}\) (remember that we normalize by \(\Vert Q\Vert =1\)). Our aim is to bound \(\alpha \) from below independently on n. We need the lemma:
Lemma 3.1
For every fixed i the total volume of the kcodimensional skeleton \(\Vert C_i^{(dk)}\Vert \) is at most \(g(d,k) \Vert C_i\Vert n^k \le g(d, k) \alpha n^{km}\).
Proof
We have started with the cubic sets \(C_i\), and for every small cube of volume \(n^{d}\) the total volume of its kcodimensional facets is equal to \(\genfrac(){0.0pt}{}{d}{k}2^k n^{dk}\). Every such kcodimensional face may split into several faces after perturbing the cubic partition; but this splitting multiplicity can also be estimated. For example, if we use the perturbation by shifting it may split into \(2^{k1}\) parts at most. In this case, by denoting \(g(d, k) = \genfrac(){0.0pt}{}{d}{k}2^{2k1}\) we obtain the required inequality. In other cases an analogous inequality holds with another \(g(d,k)\). \(\square \)
Now we do what is called in [4] “contraction in the space of cycles”. We actually use an elementary version of contraction (without using the Dold–Thom–Almgren theorem or other hard topological machinery) similar to the reasoning in [7], or in the nice review [5].
Footnotes
Notes
Acknowledgments
The author thanks Jiří Matoušek and Imre Bárány for intensive discussions and numerous useful remarks. The research is supported by the Dynasty Foundation, by ERC Advanced Research Grant No. 267195 (DISCONV), the President’s of Russian Federation grant MD352.2012.1, the Federal Program “Scientific and scientificpedagogical staff of innovative Russia” 2009–2013, and the Russian government project 11.G34.31.0053.
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