Linear polychromatic colorings of hypercube faces

A coloring of the $\ell$-dimensional faces of $Q_n$ is called $d$-polychromatic if every embedded $Q_d$ has every color on at least one face. Denote by $p^\ell(d)$ the maximum number of colors such that any $Q_n$ can be colored in this way. We provide a new lower bound on $p^\ell(d)$ for $\ell>1$.


Introduction
Denote by Q n the n-dimensional hypercube on 2 n vertices.Definition 1.1.For ℓ ≥ 0, a Q ℓ -coloring of Q n is a coloring of each of the the ℓ-dimensional faces of Q n with one of r ≥ 1 colors.For d ≥ ℓ, such a coloring is called d-polychromatic if every embedded Q d contains all r colors.
For d ≥ ℓ ≥ 1, we denote by p ℓ (d) the maximum r for which a dpolychromatic Q ℓ -coloring is possible on every hypercube Q n , for all n ≥ d.
The case ℓ = 1 was first introduced in 2007 by Alon, Krech, and Szabó in [1].They prove the following result.
Theorem ([1, Theorem 4]).For any d ≥ 1, The lower bound is done through a construction which in this paper will be called the basic construction, described in Section 2. It was then shown by Offner in 2008 that in fact this construction is sharp.
Theorem ( [4]).For any d ≥ 1, we have Alon, Krech, and Szabó also suggest in [1] the problem of examining p ℓ (d).In 2015, Ozkahya and Stanton [5] gave a direct generalization of the basic construction to prove the following.
Theorem ( [5]).For any d, ℓ ≥ 1, let 0 < r ≤ ℓ + 1 be such that r ≡ d + 1 (mod ℓ + 1).Then Henceforth, denote the right-hand side as for brevity.For ℓ = 1 this coincides with the result of [1].It is then natural to wonder whether an analog of Offner's result holds for ℓ > 1.In a few small cases it was recently shown this is not the case; Goldwasser, Lidicky, Martin, Offner, Talbot and Young prove in [3] the following result.
In the present paper we show the following more general result.
Theorem 1.2.For d ≥ 4, we have In particular, p 2 (d) > p 2 bas (d).Our construction is by a so-called linear coloring, defined at the end of Section 2. For concreteness, Table 1 lists the values of the construction for 4 ≤ d ≤ 12, as well as the bounds given by p 2 bas (d) and d+1 3 .This easily implies that unlike ℓ = 1, we have p ℓ (d) > p ℓ bas (d) for any d > ℓ > 1 (with the d = ℓ + 1 case following from p 2 (3) = 3).We state this formally as the following corollary.The rest of the paper is structured as follows.In Section 2 we present the background theory and information for the problem, and in Section 3 we prove the construction which gives the bound in Theorem 1.2.Finally in Section 4 we mention some upper bounds on the number of colors possible in a linear polychromatic coloring.The author thanks Joe Gallian for supervising the research and for suggesting the problem, as well as helpful comments on early drafts of the paper.The author would also like to acknowledge the anonymous referee for several corrections and suggestions on the paper.

Simple and linear colorings
It is conventional to refer to the vertices of Q n with n-dimensional binary strings, and to represent an embedded Q k by writing * in the corresponding coordinates.For example, in Q 8 the embedded Q 2 whose four vertices are 01000011, 01001011, 01100011, 01101011, is typically represented by 01 * 0 * 011.
We say that a Q ℓ -coloring is simple if the color of each Q d depends only on the number of 1's in the d + 1 regions (possibly empty) delimited by the * 's.For example, in a simple 2-polychromatic coloring of Q 7 , the faces 01 * 0 * 011 and 10 * 0 * 101 would be assigned the same color.
The following generalization of [1, Claim 10] (present also as [3, Lemma 18] and [5, Claim 6]) shows that in fact it suffices to only consider simple colorings.The proof is a nice application of the Ramsey theorem.
Thus for the purposes of coloring, we can consider an embedded Q k in Q n as a sequence of nonnegative integers (a 0 , a 1 , . . ., a k ) such that a i denotes the number of 1's between the ith and (i + 1)st star.For example, 01 * 0 * 011 can be identified with (1, 0, 2).In light of this a Q ℓ -coloring with colors from a set S can be thought of as a function χ : Z ℓ+1 ≥0 ։ S. We can now motivate the so-called basic colorings as follows.
Definition 2.2.For n ≥ d ≥ ℓ ≥ 1, choose positive integers m 0 , m 1 , . . ., m ℓ with sum d + 1 and consider the coloring We call any coloring of this form a basic coloring.
We claim this gives a basic 14-polychromatic Q 2 -coloring Consider an embedded Q 14 in some Q n , which can be thought of as a sequence of 14 stars.Select the 5th and 9th star as follows, and denote the remaining bits by ε 1 , . . ., ε 12 , as shown below.* * * * * * * * * * * This gives 2 12 = 4096 choices of Q 2 faces in our embedded Q 14 .We claim that all colors are present among just these faces.Let x, y, z denote the number of 1's from the ambient Q n present in the three regions cut out by the boxed stars.Then, we wish to show that achieves all colors, which is obvious since More generally, as shown in [5, Theorem 1], every basic coloring is indeed seen to be d-polychromatic.The lower bound p ℓ bas (d) now follows by taking the m i such that |m i − m j | ≤ 1 for all 1 ≤ i < j ≤ ℓ.Definition 2.4.More generally, a linear coloring is one where the colors are selected from some (finite) abelian group Z, and which is induced by an additive map χ : Z ℓ+1 ≥0 ։ Z.

A family of linear colorings
We now exhibit a family of linear d-polychromatic Q ℓ -colorings.
Proof.We begin by addressing the first case Fix an embedding Q d , which as usual we think of as a sequence of d stars embedded in an ambient string of 1's and 0's.We can represent this with the diagram where x i denotes the number of 1's in the region delimited by those two stars.
First, consider the family of squares cut out by the star pattern where we consider the squares formed when all the bits other than the tth and 2tth bit are assigned a particular value.For example, the square Now suppose we vary the choice of assigned bits.First consider the last n − 1 stars.Since {0, 1, . . ., n − 1} covers all residues modulo n, we see that the second coordinate is arbitrary, even regardless of the choices of the first 2(t − 1) stars.Moreover, the first coordinate doesn't depend on the choice of these last n − 1 stars.
So we focus on the first coordinate.Let 0 ≤ u ≤ t − 1 and 0 ≤ v ≤ t − 1 be the number of 1's we select in the first and second regions, respectively.(Thus the first coordinate receives color X + u − tv.)The values of u − tv (modulo m) are given in the table Thus, we see that we achieve exactly the colors with first coordinate in the set X +{0, 1, . . ., t−1, t+1, t+2, . . ., t 2 } so the colors not present are exactly those whose first coordinate is Next, consider the family * t * * t−2 * * n−1 and this time define which is the first coordinate of the analogous all-zero color.Again, consider varying the choice of assigned bits, this time with u ∈ {0, 1, . . ., t} and v ∈ {0, . . ., t − 2}.The values of u − tv are given in the table So by the same argument as in the previous case, the colors not present are exactly those whose first coordinate is in the set If Y − X / ∈ {1, 2, . . ., t} then we are now done.Let δ = Y − X and henceforth assume Y −X ∈ {1, 2, . . ., t}.We denote by k = X +t = Y +t+δ, and call any color of the form (k, •) a "critical color."We wish to show all n critical colors are present on some other face.
We consider the two families * t−1 * * t * * n−2 * t * * t−1 * * n−2 which we will call the "first" family and the "second" family.Let C = −tx 2t (mod m).As before, the all-zero squares in these families receive the colors (X + C, S) ∈ Z and (Y + C, S) ∈ Z, respectively.
Define u and v as before and now let 0 ≤ w ≤ n − 2 denote the number of 1's in the rightmost region.Again, we can exhibit two tables for u and v defined as before: for the first family we obtain a table Proof of Theorem 1.2.In Theorem 3.1, take the following choices of parameters:

Upper bounds
We do not have at present any upper bound for p 2 (Q d ) other than the simple d+1 3 bound.In this section we briefly mention an upper bound for the number of colors in a linear d-polychromatic coloring.
Specifically, we use the geometry of numbers to prove the following.Proof.Let N = |Z|.Extend χ to a map Z 3 ։ Z of abelian groups.Then consider Z 3 as a tetrahedral lattice Λ 0 in R 3 .In this case, the kernel of χ is a lattice Λ of index N in Z 3 .
Let n = d − 2. Now if we consider the coloring of Q d itself by χ (or really any embedding of Q d into Q N with all ambient bits zero), we see that the colors present are precisely those χ(x, y, z) where x+y +z ≤ n, x, y, z ∈ Z ≥0 .Thus we obtain a regular tetrahedron T of side length n in which all colors are present.
On the other hand suppose that Λ contains a nonzero vector v which fits inside a regular tetrahedron of side length s > 0. Therefore for any p ∈ Z 3 , χ assigns the same color to both p and p + v.In particular, this implies all the colors are present in a frustum of T with height s layers; this gives a bound of ( 4) Now let c be the length of the shortest nonzero vector in Λ.Then since a tetrahedron has height equal to 2/3 times its side length, we may take (5) s = 3/2c .
Next we bring in the theory of sphere packing.Observe that if we construct spheres of diameter c centered at each point in Λ, then we have obtained a packing of spheres in R 3 .We have det(Λ) = N det(Λ 0 ), but Λ 0 is known to be an optimal packing of 3-spheres (see e.g.[2]), and so from this we deduce that (6) 0 < c ≤ 3 √ N .
Collating (4), ( 5), (6) together we deduce the inequality c ≤ It would be interesting if any stronger upper bounds could be proven for polychromatic colorings, linear or otherwise.