Fix $k \in \mathbb{N}, 0 < δ< 1$. We study how large $N$ must be so that every $δ$-dense subset $\mathcal{D} \subset \{0,1\}^N$ (meaning $|\mathcal{D}|\geq δ2^N$) contains the image of a metric embedding $f: \{0,1\}^k \to \mathcal{D}$. We study $3$ variants: For a $(1+\varepsilon)$-bi-Lipschitz map $f$ for a fixed $\varepsilon>0$, we show that $N = O(\varepsilon^{-2}\log(1/δ) k^3)$. For an isometric map $f$ with arbitrary rescaling (i.e. undistorted), we show that $N = \log(1/δ) e^{Ω(k)}$. For an isometric map $f$ with bounded rescaling we show that $N = \exp{[\log(1/δ)e^{Θ(k)}]}$.
Regarding the path space, we prove the density analog of a coloring theorem of Rödl--Sales. We give bounds for $(1+\varepsilon)$-bi-Lipschitz embeddings of the path $[k] = \{1,...,k\}$ into dense subsets of the path $[N] = \{1,...,N\}$, improving a bound of Dumitrescu. We prove similar bounds for the binary tree space, using the tree replicas theorem of Pach--Solymosi--Tardos.
As a geometric application we obtain a non-positive Alexandrov curvature counterpart to the work of Bartal--Linial--Mendel--Naor on the nonlinear Dvoretzky problem who showed that any $\mathcal{D} \subset \{0,1\}^N$ that embeds with bi-Lipschitz distortion $<α$ into a metric space of non-negative Alexandrov curvature must be small, namely, necessarily $|\mathcal{D}| \lesssim 2^{N(1-Ω(α^{-2}))}$. We prove that for every $N\gtrsim α^{6}\ge 1$, any $\mathcal{D} \subset \{0,1\}^N$ that embeds with distortion $<α$ into some metric space of non-positive Alexandrov curvature must satisfy $|\mathcal{D}| \lesssim 2^{N(1-Ω(α^{-4}))}$ via an approach which is entirely different from that of Bartal--Linial--Mendel--Naor. We also show that nontrivial metric type and non-universality are preserved by taking finite unions of subspaces.
An \emph{$n$-cube antichain} is a subset of the unit $n$-cube $[0,1]^n$ that does not contain two elements $\mathbf{x}=(x_1, x_2,\ldots, x_n)$ and $\mathbf{y}=(y_1, y_2,\ldots, y_n)$ satisfying $x_i\le y_i$ for all $i\in \{1,\ldots,n\}$. Using a chain partition of an adequate finite poset we show that the Hausdorff dimension of an $n$-cube antichain is at most $n-1$.We conjecture that the $(n-1)$-dimensional Hausdorff measure of an $n$-cube antichain is at most $n$ times the Hausdorff measure of a facet of the unit $n$-cube and we verify this conjecture for $n=2$ as well as under the assumptio…
The edge isoperimetric inequality in the discrete cube specifies, for each pair of integers $m$ and $n$, the minimum size $g_n(m)$ of the edge boundary of an $m$-element subset of $\{0,1\}^{n}$; the extremal families (up to automorphisms of the discrete cube) are initial segments of the lexicographic ordering on $\{0,1\}^n$. We show that for any $m$-element subset $\mathcal{F} \subset \{0,1\}^n$ and any integer $l$, if the edge boundary of $\mathcal{F}$ has size at most $g_n(m)+l$, then there exists an extremal family $\mathcal{G} \subset \{0,1\}^n$ such that $|\mathcal{F} Δ\mathcal{G}| \leq C…
We provide a formula for the number of edges of the Hasse diagram of the independent subsets of the h-th power of a path ordered by inclusion. For h=1 such a value is the number of edges of a Fibonacci cube. We show that, in general, the number of edges of the diagram is obtained by convolution of a Fibonacci-like sequence with itself.
This paper consists of two independent, but related parts. In the first part we show how to use symbolic computation to derive explicit expressions for the first few moments of the number of implicants that a random Boolean function has, or equivalently the number of fixed-dimensional subcubes contained in a random subset of the $n$-dimensional cube. These explicit expressions suggest, but do not prove, that these random variables are always asymptotically normal. The second part presents a full, human-generated proof, of this asymptotic normality, first proved by Urszula Konieczna.
A family of subsets of $\{1,2,\ldots,n\}$ is said to be {\em antipodal} if it is closed under taking complements. We prove a best-possible isoperimetric inequality for antipodal families of subsets of $\{1,2,\ldots,n\}$. Our inequality implies that for any $k \in \mathbb{N}$, among all such families of size $2^k$, a family consisting of the union of a $(k-1)$-dimensional subcube and its antipode has the smallest possible edge boundary.
In the Boolean lattice, Sperner's, Erdős's, Kleitman's and Samotij's theorems state that families that do not contain many chains must have a very specific layered structure. We show that if instead of $\mathbb{Z}_2^n$ we work in $\mathbb{Z}_{2^n}$, several analogous statements hold if one replaces the word $k$-chain by projective cube of dimension $2^{k-1}$. We say that $B_d$ is a projective cube of dimension $d$ if there are numbers $a_1, a_2, \ldots, a_d$ such that $$B_d = \left\{\sum_{i\in I} a_i \bigg\rvert \emptyset \neq I\subseteq [d]\right\}.$$ As an analog of Sperner's and Erdős's the…
We examine packing of $n$ congruent spheres in a cube when $n$ is close but less than the number of spheres in a regular cubic close-packed (ccp) arrangement of $\lceil p^{3}/2\rceil$ spheres. For this family of packings, the previous best-known arrangements were usually derived from a ccp by omission of a certain number of spheres without changing the initial structure. In this paper, we show that better arrangements exist for all $n\leq\lceil p^{3}/2\rceil-2$. We introduce an optimization method to reveal improvements of these packings, and present many new improvements for $n\leq4629$.
A subset $M \subset \textbf{R}^3$ is called a \emph{basic subset}, if for any funciton $f \colon M \to \textbf{R}$ there exist such functions $f_1; f_2; f_3 \colon \textbf{R} \to \textbf{R}$ that $f(x_1, x_2, x_3) = f_1(x_1) + f_2(x_2) + f_3(x_3)$ for each point $(x_1, x_2, x_3)\in M$. In this article we prove a criterion for a basic subset for some specific subsets in terms of some graph properties. We also introduce several constructions for mimimal non-basic subsets. The article is written in Russian.
Let Q_n be the graph of n times n times n cube with all non-decreasing diagonals (including the facial ones) in its constituent unit cubes. Suppose that a subset S of V(Q_n) separates the left side of the cube from the right side. We show that S induces a subgraph of tree-width at least n/sqrt{18}-1. We use a generalization of this claim to prove that the vertex set of Q_n cannot be partitioned to two parts, each of them inducing a subgraph of bounded tree-width.
We show that a set $A \subset \{0,1\}^{n}$ with edge-boundary of size at most $|A| (\log_{2}(2^{n}/|A|) + ε)$ can be made into a subcube by at most $(2 ε/\log_{2}(1/ε))|A|$ additions and deletions, provided $ε$ is less than an absolute constant. We deduce that if $A \subset \{0,1\}^{n}$ has size $2^{t}$ for some $t \in \mathbb{N}$, and cannot be made into a subcube by fewer than $δ|A|$ additions and deletions, then its edge-boundary has size at least $|A| \log_{2}(2^{n}/|A|) + |A| δ\log_{2}(1/δ) = 2^{t}(n-t+δ\log_{2}(1/δ))$, provided $δ$ is less than an absolute constant. This is sharp wheneve…
Let ${[n] \choose k}$ and ${[n] \choose l}$ $( k > l ) $ where $[n] = \{1,2,3,...,n\}$ denote the family of all $k$-element subsets and $l$-element subsets of $[n]$ respectively. Define a bipartite graph $G_{k,l} = ({[n] \choose k},{[n] \choose l},E)$ such that two vertices $S\, ε\,{[n] \choose k} $ and $T\, ε\,{[n] \choose l} $ are adjacent if and only if $T \subset S$. In this paper, we give an upper bound for the domination number of graph $G_{k,2}$ for $k > \lceil \frac{n}{2} \rceil$ and exact value for $k=n-1$.
A non-empty subset A of X = X_1 x...x X_d is a (proper) box if A = A_1 x...x A_d and A_i is a (proper) subset of X_i for each i. Suppose that for each pair of boxes A, B and each i, one can only know which of the three states takes place: A_i = B_i, A_i = X_i\setminus B_i, A_i is different from each of the sets B_i, X_i\setminus B_i. Let F and G be two systems of disjoint boxes. Can one decide whether their unions are equal? In general, the answer is no, but as is shown in the paper, it is yes if both systems consist of pairwise dichotomous boxes. Several criteria that enable to compare such s…
We investigate gaps of $n$-term arithmetic progressions $x, x+y, \ldots, x+(n-1)y$ inside a positive measure subset $A$ of the unit cube $[0,1]^d$. If lengths of their gaps $y$ are evaluated in the $\ell^p$-norm for any $p$ other than $1, 2, \ldots, n-1$, and $\infty$, and if the dimension $d$ is large enough, then we show that the numbers $\|y\|_{\ell^p}$ attain all values from an interval, the length of which depends only on $n$, $p$, $d$, and the measure of $A$. Known counterexamples prevent generalizations of this result to the remaining values of the exponent $p$. We also give an explicit…
Consider all $k$-element subsets and $\ell$-element subsets $(k>\ell )$ of an $n$-element set as vertices of a bipartite graph. Two vertices are adjacent if the corresponding $\ell$-element set is a subset of the corresponding $k$-element set. Let $G_{k,\ell}$ denote this graph. The domination number of $G_{k,1}$ was exactly determined by Badakhshian, Katona and Tuza. A conjecture was also stated there on the asymptotic value ($n$ tending to infinity) of the domination number of $G_{k,2}$. Here we prove the conjecture, determining the asymptotic value of the domination number $γ(G_{k,2})={k+3\…
We prove the following stability version of the edge isoperimetric inequality for the cube: any subset of the cube with average boundary degree within $K$ of the minimum possible is $\varepsilon $-close to a union of $L$ disjoint cubes, where $L \leq L(K,\varepsilon )$ is independent of the dimension. This extends a stability result of Ellis, and can viewed as a dimension-free version of Friedgut's junta theorem.
For an integer $n\geq 2$, the triangular graph has vertex set the $2$-subsets of $\{1,\ldots,n\}$ and edge set the pairs of $2$-subsets intersecting at one point. Such graphs are known to be halved graphs of bipartite rectagraphs, which are connected triangle-free graphs in which every $2$-path lies in a unique quadrangle. We refine this result and provide a characterisation of connected locally triangular graphs as halved graphs of normal quotients of $n$-cubes. To do so, we study a parameter that generalises the concept of minimum distance for a binary linear code to arbitrary automorphism g…
In this note we establish a Ramsey-type result for certain subsets of the $n$-dimensional cube. This can then be applied to obtain reasonable bounds on various related structures, such as (partial) Hales-Jewett lines for alphabets of sized 3 and 4, Hilbert cubes in sets of real numbers with small sumsets, "corner" in the integer lattice in the plane, and 3-term geometric progressions in integers.
A \emph{chain} in the unit $n$-cube is a set $C\subset [0,1]^n$ such that for every $\mathbf{x}=(x_1,\ldots,x_n)$ and $\mathbf{y}=(y_1,\ldots,y_n)$ in $C$ we either have $x_i\le y_i$ for all $i\in [n]$, or $x_i\ge y_i$ for all $i\in [n]$. We consider subsets, $A$, of the unit $n$-cube $[0,1]^n$ that satisfy \[ \text{card}(A \cap C) \le k, \, \text{ for all chains } \, C \subset [0,1]^n \, , \] where $k$ is a fixed positive integer. We refer to such a set $A$ as a $k$-antichain. We show that the $(n-1)$-dimensional Hausdorff measure of a $k$-antichain in $[0,1]^n$ is at most $kn$ and that the b…
For each $1\leq i \le n$, let $k_i\geq 1$ and let $Δ_i$ be a set of vertices of a non-degenerate simplex of $k_i+1$ points in $\mathbb{R}^{k_i+1}$. If $A\subseteq [0,1]^{k_1+1}\times \cdots \times [0,1]^{k_n+1}$ is a Lebesgue measurable set of measure at least $δ$, we show that there exists an interval $I=I(Δ_1,\ldots, Δ_n,A)$ of length at least $\exp(-δ^{-C(Δ_1,\ldots, Δ_n)})$ such that for each $λ\in I$, the set $A$ contains $Δ'_1\times \cdots \times Δ'_n$, where each $Δ_i'$ is an isometric copy of $λΔ_i$. This is a quantitative improvement of a result by Lyall and Magyar. Our proof relies o…
We show that in any subset of the vertices of $n$-dimensional cube that contains at least $2^{n-1}+1$ vertices ($n\geq 4$), there are four vertices that induce a claw, or there are eight vertices that induce the cycle of length eight.
The Vapnik-Chervonenkis (VC) dimension of a collection of subsets of a set is an important combinatorial concept in settings such as discrete geometry and machine learning. In this paper we prove that the VC dimension of the family of $d$-dimensional cubes in $\mathbb R^d$ is $\lfloor(3d+1)/2\rfloor$.
The logarithmic Sobolev inequality for the Hamming cube {0,1}^n states that for any real-valued function f on the cube holds E(f,f) \ge 2 Ent(f^2), where E(f,f) is the appropriate Dirichlet form (also known as "sum of influences"). We show that the constant C = 2 at the right hand side of this inequality can be replaced by a function C(rho) depending on rho = Ent(f^2) / (n Ef^2). The function C is an increasing convex function taking [0,log 2] to [2, 2/log 2]. We present some applications of this modified inequality. In particular, it is used to obtain a discrete version of the Faber-Krahn ine…
We consider the realization space of the $d$-dimensional cube, and show that any two realizations are connected by a finite sequence of projective transformations and normal transformations. We use this fact to define an analog of the connected sum construction for cubical $d$-polytopes, and apply this construction to certain cubical $d$-polytopes to conclude that the rays spanned by $f$-vectors of cubical $d$-polytopes are dense in Adin's cone. The connectivity result on cubes extends to any product of simplices, and further, it shows the respective realization spaces are contractible.
We show upper bounds on the maximal dimension $d$ of Hilbert cubes $H=a_0+\{0,a_1\}+\cdots + \{0, a_d\}\subset S \cap [1, N]$ in several sets $S$ of arithmetic interest such as the squares, powerful numbers and pure powers.
A family ${\mathcal A} \subset {\mathcal P} [n]$ is said to be an antichain if $A \not \subset B$ for all distinct $A,B \in {\mathcal A}$. A classic result of Sperner shows that such families satisfy $|{\mathcal A}| \leq \binom {n}{\lfloor n/2\rfloor}$, which is easily seen to be best possible. One can view the antichain condition as a restriction on the intersection sizes between sets in different layers of ${\mathcal P} [n]$. More generally one can ask, given a collection of intersection restrictions between the layers, how large can families respecting these restrictions be? Answering a que…
Consider the metric space $\mathcal{C}$ consisting of the $n$-dimensional Boolean cube equipped with the Hamming distance. A weak isometry of $\mathcal{C}$ is a permutation of $\mathcal{C}$ preserving a given subset of Hamming distances. In \cite{Krasin} Krasin showed that in most cases preserving a single Hamming distance forces a weak isometry to be an isometry. In this article we study those weak isometries that are not automatically an isometry, providing a complete classification of weak isometries of $\mathcal{C}$.
A toric cube is a subset of the standard cube defined by binomial inequalities. These basic semialgebraic sets are precisely the images of standard cubes under monomial maps. We study toric cubes from the perspective of topological combinatorics. Explicit decompositions as CW-complexes are constructed. Their open cells are interiors of toric cubes and their boundaries are subcomplexes. The motivating example of a toric cube is the edge-product space in phylogenetics, and our work generalizes results known for that space.
We find the asymptotic behavior of the Steiner k-diameter of the $n$-cube if $k$ is large. Our main contribution is the lower bound, which utilizes the probabilistic method.
Let k_1,...,k_d be positive integers, and D be a subset of [k_1]x...x[k_d], whose complement can be decomposed into disjoint sets of the form {x_1}x...x{x_{s-1}}x[k_s]x{x_{s+1}}x...x{x_d}. We conjecture that the number of elements of D can be represented as a linear combination of the numbers k_1,..., k_d with non-negative integer coefficients. A connexion of this conjecture with the structure of periodical cube tilings is revealed.
Let $Q_n$ be the poset that consists of all subsets of a fixed $n$-element set, ordered by set inclusion. The poset cube Ramsey number $R(Q_n,Q_n)$ is defined as the least $m$ such that any 2-coloring of the elements of $Q_m$ admits a monochromatic copy of $Q_n$. The trivial lower bound $R(Q_n,Q_n)\ge 2n$ was improved by Cox and Stolee, who showed $R(Q_n,Q_n)\ge 2n+1$ for $3\le n\le 8$ and $n\ge 13$ using a probabilistic existence proof. In this paper, we provide an explicit construction that establishes $R(Q_n,Q_n)\ge 2n+1$ for all $n\ge 3$. The best known upper bound, due to Lu and Thompson,…
We examine the behavior of the number of $k$-term arithmetic progressions in a random subset of $\mathbb{Z}/n\mathbb{Z}$. We prove that if a set is chosen by including each element of $\mathbb{Z}/n\mathbb{Z}$ independently with constant probability $p$, then the resulting distribution of $k$-term arithmetic progressions in that set, while obeying a central limit theorem, does not obey a local central limit theorem. The methods involve decomposing the random variable into homogeneous degree $d$ polynomials with respect to the Walsh/Fourier basis. Proving a suitable multivariate central limit th…
Let $\mathcal S$ be a family of subsets of a set $X$ of cardinality $m$ and $\text{VC-dim}(\mathcal S)$ be the Vapnik-Chervonenkis dimension of $\mathcal S$. Haussler, Littlestone, and Warmuth (Inf. Comput., 1994) proved that if $G_1(\mathcal S)=(V,E)$ is the subgraph of the hypercube $Q_m$ induced by $\mathcal S$ (called the 1-inclusion graph of $\mathcal S$), then $\frac{|E|}{|V|}\le \text{VC-dim}({\mathcal S})$. Haussler (J. Combin. Th. A, 1995) presented an elegant proof of this inequality using the shifting operation. In this note, we adapt the shifting technique to prove that if $\mathca…
In \cite{FKW} Katznelson and Weiss establish that all sufficiently large distances can always be attained between pairs of points from any given measurable subset of $\mathbb{R}^2$ of positive upper (Banach) density. A second proof of this result, as well as a stronger "pinned variant", was given by Bourgain in \cite{B} using Fourier analytic methods. In \cite{M1} the second author adapted Bourgain's Fourier analytic approach to established a result analogous to that of Katznelson and Weiss for subsets $\mathbb{Z}^d$ provided $d\geq 5$. We present a new direct proof of this discrete distance s…
