Combinatorial Cube Research Articles

Sums of triples in Abelian groups

AbstractMotivated by a problem in additive Ramsey theory, we extend Todorčević's partitions of three‐dimensional combinatorial cubes to handle additional three‐dimensional objects. As a corollary, we get that if the continuum hypothesis fails, then for every Abelian group G of size ℵ2, there exists a coloring such that for every uncountable and every integer k, there are three distinct elements of X such that .

Remixed Eulerian numbers

Abstract Remixed Eulerian numbers are a polynomial q-deformation of Postnikov’s mixed Eulerian numbers. They arose naturally in previous work by the authors concerning the permutahedral variety and subsume well-known families of polynomials such as q-binomial coefficients and Garsia–Remmel’s q-hit numbers. We study their combinatorics in more depth. As polynomials in q, they are shown to be symmetric and unimodal. By interpreting them as computing success probabilities in a simple probabilistic process we arrive at a combinatorial interpretation involving weighted trees. By decomposing the permutahedron into certain combinatorial cubes, we obtain a second combinatorial interpretation. At $q=1$ , the former recovers Postnikov’s interpretation whereas the latter recovers Liu’s interpretation, both of which were obtained via methods different from ours.

On sums and products of combinatorial cubes

We obtain several lower bounds on the product set of combinatorial cubes, as well as some non–trivial upper estimates for the multiplicative energy of such sets.

Degeneracy of the Intersection of Three Quadrics

We prove that the 8-point algorithm always fails to reconstruct a unique fundamental matrix F independent on the camera positions, when its inputs are image point configurations that are perspective projections of the intersection of three quadrics in \(\mathbb {P}^3\). This generalizes a multiple results of the degeneracies of the 8-point algorithm. We give an algorithm that improves the 7- and 8-point algorithm in such a pathological situation. Additionally, we analyze the regions of focal point positions where a reconstruction of F is possible at all, when the world points are the vertices of a combinatorial cube in \(\mathbb {R}^3\).

The volume of the Lambert cube in spherical space

The Lambert cube Q(α, β, γ) is one of the simplest polyhedra. By definition, this is a combinatorial cube with dihedral angles α, β, and γ at three noncoplanar edges and with right angles at all other edges. The volume of the Lambert cube in hyperbolic space was obtained by R. Kellerhals (1989) in terms of the Lobachevskii function Λ(x). In the present paper, we find the volume of the Lambert cube in spherical space. It is expressed in terms of the function $$ \delta (\alpha ,\theta ) = \int_\theta ^{\pi /2} {\log (1 - \cos 2\alpha \cos 2\tau )} \frac{{d\tau }} {{\cos 2\tau }}, $$ which can be regarded as the spherical analog of the function $$ \Delta (\alpha ,\theta ) = \Lambda (\alpha + \theta ) - \Lambda (\alpha - \theta ). $$

Combinatorial cube packings in the cube and the torus

We consider sequential random packing of cubes z + [ 0 , 1 ] n with z ∈ 1 N Z n into the cube [ 0 , 2 ] n and the torus R n / 2 Z n as N → ∞ . In the cube case, [ 0 , 2 ] n as N → ∞ , the random cube packings thus obtained are reduced to a single cube with probability 1 − O ( 1 N ) . In the torus case, the situation is different: for n ≤ 2 , sequential random cube packing yields cube tilings, but for n ≥ 3 with strictly positive probability, one obtains non-extensible cube packings. So, we introduce the notion of combinatorial cube packing, which instead of depending on N depends on some parameters. We use them to derive an expansion of the packing density in powers of 1 N . The explicit computation is done in the cube case. In the torus case, the situation is more complicated and we restrict ourselves to the case N → ∞ of strictly positive probability. We prove the following results for torus combinatorial cube packings: • We give a general Cartesian product construction. • We prove that the number of parameters is at least n ( n + 1 ) 2 and we conjecture it to be at most 2 n − 1 . • We prove that cube packings with at least 2 n − 3 cubes are extensible. • We find the minimal number of cubes in non-extensible cube packings for n odd and n ≤ 6 .

A Large Deviations Approach to the Geometry of Random Polytopes

The aim of this article is to present a general “large deviations approach” to the geometry of polytopes spanned by random points with independent coordinates. The origin of our work is in the study of the structure of ±1-polytopes, the convex hulls of subsets of the combinatorial cube . Understanding the complexity of this class of polytopes is important for the “polyhedral combinatorics” approach to combinatorial optimization, and was put forward by Ziegler in [20]. Many natural questions regarding the behaviour of ±1-polytopes in high dimensions are open, since, for many important geometric parameters, low-dimensional intuition does not help to identify the extremal ±1-polytopes. The study of random ±1-polytopes sheds light to some of these questions, the main reason being that random behaviour is often the extremal one.

Lipschitz representations of subsets of the cube

We show that for any class of uniformly bounded functions H H with a reasonable combinatorial dimension, the vast majority of small subsets of the n n -dimensional combinatorial cube cannot be represented as a Lipschitz image of a subset of H H , unless the Lipschitz constant is very large. We apply this result to the case when H H consists of linear functionals of norm at most one on a Hilbert space.

On Combinatorial Cubes

A sequence of positive integers with positive lower density contains a Hilbert (or combinatorial) cube size c log log n up to n. We prove this bound is sharp for some thinner sequence.

Lambert Cubes Generating Discrete Reflection Groups

A Lambert cube Q(α,β,γ) is a combinatorial cube with dihedral angles α, β, and γ assigned to the three mutually noncomplanar edges and right angles at the remaining edges. In this paper, we classify the Lambert cubes in S3, \(\mathbb{E}^3 \), and \(\mathbb{H}^3 \) such that the group GQ generated by the reflections with respect to the faces of a cube Q is discrete.

The Random‐Facet simplex algorithm on combinatorial cubes

AbstractThe RANDOM‐FACET algorithm is a randomized variant of the simplex method which is known to solve any linear program with n variables and m constraints using an expected number of pivot steps which is subexponential in both n and m. This is the theoretically fastest simplex algorithm known to date if m ≈ n; it provably beats most of the classical deterministic variants which require exp(Ω(n)) pivot steps in the worst case. RANDOM‐FACET has independently been discovered and analyzed ten years ago by Kalai as a variant of the primal simplex method, and by Matous̆ek, Sharir, and Welzl in a dual form. The essential ideas and results connected to RANDOM‐FACET can be presented in a particularly simple and instructive way for the case of linear programs over combinatorial n‐cubes. I derive an explicit upper bound of on the expected number of pivot steps in this case, using a new technique of “fingerprinting” pivot steps. This bound also holds for generalized linear programs, similar flavors of which have been introduced and studied by several researchers. I then review an interesting class of generalized linear programs, due to Matous̆ek, showing that RANDOM‐FACET may indeed require an expected number of $\exp \bigl(\Omega\bigl(\sqrt n\bigr)\bigr)$ pivot steps in the worst case. The main new result of the paper is a proof that all actual linear programs in Matous̆ek's class are solved by RANDOM‐FACET with an expected polynomial number of $O \bigl(n^2 \bigr)$ pivot steps. This proof exploits a combinatorial property of linear programming which has only recently been discovered by Holt and Klee. The result establishes the first scenario in which an algorithm that works for generalized linear programs “recognizes” proper linear programs. Thus, despite Matous̆ek's worst‐case result, the question remains open whether RANDOM‐FACET (or any other simplex variant) is a polynomial‐time algorithm for linear programming. Finally, I briefly discuss extensions of the combinatorial cube results to the general case. © 2002 Wiley Periodicals, Inc. Random Struct. Alg., 20:353–381, 2002

The almost simple cubical polytopes

A cubical polytope is a convex polytope all of whose facets are combinatorial cubes. A d-polytope P is called almost simple if, in the graph of P, each vertex of P is d-valent or ( d + 1)-valent. We give a complete enumeration of all the almost simple cubical d-polytopes for d ⩾ 4, which is even valid for almost simple cubical ( d − 1)-spheres. This provides a complete enumeration of all the cubical d-polytopes with up to 2 d+1 vertices for d ⩾ 4. With a single exception, these are precisely the d-polytopes which can be embedded into a ( d + 1)-cube.

Cubical d -Polytopes with Few Vertices for d >4

A cubical polytope is a convex polytope all of whose facets are combinatorial cubes. A d-polytope P is called almost simple if, in the graph of P, each vertex of P is d-valent or (d+1)-valent. We show that, for d>4, all but one cubicald -polytopes with up to 2d+1 vertices are almost simple. This provides a complete enumeration of all the cubical d-polytopes with up to 2d+1 vertices, for d>4.

The cubicald-polytopes with fewer than 2 d+1 vertices

A cubical polytope is a convex polytope of which every facet is a combinatorial cube. We give here a complete enumeration of all the cubicald-polytopes with fewer than 2d+1 vertices, ford≥4.

Some partitions of three-dimensional combinatorial cubes

Some partitions of three-dimensional combinatorial cubes

Gaps in the numbers of vertices of cubical polytopes, I

A cubical polytope is a convex polytope of which very facet is a combinatorial cube. We ask for the numbers which occur as vertex numbers ofd-dimensional cubical polytopes, and we show, as a first step, that every cubicald-polytope for evend?4 has an even number of vertices.

On a glass of abstract polytopes constructed from binary codes

Certain abstract polytopes are constructed from generalized combinatorial cubes by taking quotients with respect to elementary Abelian subgroups (linear codes) of the automorphism group.

Some remarks on natural orders for combinatorial cubes

In this note, a new combinatorial interpretation of the natural linear orders on combinatorial cubes over finite sets A is given. This allows us to simplify the proof of the canonizing ordering theorem for combinatorial cubes [7] considerably. Moreover, we obtain an extension of this theorem in case |A| = 2, which can be interpreted as a characterization theorem for natural partial orders on Boolean lattices.

A sparse Graham-Rothschild theorem

The main result of this paper is a sparse version of the Graham-Rothschild partition theorem for n n -parameter sets [R. L. Graham and B. L. Rothschild, Ramsey’s theorem for n n -parameter sets, Trans. Amer. Math. Soc. 159 (1971), 257-292]. In particular, a sparse version of Hales-Jewett’s theorem is proved. We give several applications, e.g., for arithmetic progressions and finite sums of integers, confirming conjectures of J. Spencer and of J. Nešetřil and V. Rödl. We also consider graphs defined on parameter sets and prove a sparse and restricted induced partition theorem for such graphs, extending results from [H. J. Prömel, Induced partition properties of combinatorial cubes, J. Combin. Theory Ser. A 39 (1985), 177-208] and [P. Frankl, R. L. Graham, and V. Rödl, Induced restricted Ramsey theorems for spaces, J. Combin. Theory Ser. A 44 (1987), 120-128].

Induced partition properties of combinatorial cubes

An induced version of the partition theorem for parameter-sets of R. L. Graham and B. L. Rothschild ( Trans. Amer. Math. Soc. 159 (1971), 257–291) is proven. This theorem generalizes the Graham-Rothschild theorem in the same way as the partition theorem for finite hypergraphs (F. G. Abramson and L. A. Harrington, J. Symblic Logic 43 (1978), 572–600 and J. Nešetřil and V. Rödl; J. Combin. Theory Ser. A 22 (1977), 289–312; 34 (1983), 183–201) generalizes Ramsey's theorem. Some applications are given, e.g., an induced version of the Rado-Folkman-Sanders theorem and an induced version of the partition theorem for finite Boolean lattices. Also it turns out that the partition theorem for finite hypergraphs is an easy consequence of the induced partition theorem for parameter-sets.