Related papers: Uniform chain decompositions and applications
We prove that Boolean matrices with bounded $\gamma_2$-norm or bounded normalized trace norm must contain a linear-sized all-ones or all-zeros submatrix, verifying a conjecture of Hambardzumyan, Hatami, and Hatami. We also present further…
We present a short proof that every maximal family of weakly separated subsets of $[n]$ of cardinality between $[a,b]$ have the same size. Our proof is direct and only uses elementary combinatorics of lattice paths.
A labeling scheme for nearest common ancestors assigns a distinct binary string, called the label, to every node of a tree, so that given the labels of two nodes (and no further information about the topology of the tree) we can compute the…
We show that there is some absolute constant $c>0$, such that for any union-closed family $\mathcal{F} \subseteq 2^{[n]}$, if \mbox{$|\mathcal{F}| \geq (\frac{1}{2}-c)2^n$}, then there is some element $i \in [n]$ that appears in at least…
The crown $\Oh_{2t}$ is a height-2 poset whose Hasse diagram is a cycle of length $2t$. A family $\F$ of subsets of $[n]:=\{1,2..., n\}$ is {\em $\Oh_{2t}$-free} if $\Oh_{2t}$ is not a weak subposet of $(\F,\subseteq)$. Let…
Let $F$ be a family of subsets of $\{1,\ldots,n\}$. We say that $F$ is $P$-free if the inclusion order on $F$ does not contain $P$ as an induced subposet. The \emph{Tur\'an function} of $P$, denoted $\pi^*(n,P)$, is the maximum size of a…
The current paper investigates the bounded distance decoding (BDD) problem for ensembles of lattices whose generator matrices have sub-Gaussian entries. We first prove that, for these ensembles the BDD problem is NP-hard in the worst case.…
Since a plurality of processors in a distributed computer system working in parallel, to ensure the fault tolerance and stability of the network is an important issue in distributed systems. As the topology of the distributed network can be…
The notion of a difference hierarchy, first introduced by Hausdorff, plays an important role in many areas of mathematics, logic and theoretical computer science such as descriptive set theory, complexity theory, and the theory of regular…
An induced subposet $(P_2,\le_2)$ of a poset $(P_1,\le_1)$ is a subset of $P_1$ such that for every two $X,Y\in P_2$, $X\le_2 Y$ if and only if $X\le_1 Y$. The Boolean lattice $Q_n$ of dimension $n$ is the poset consisting of all subsets of…
How large can a family \cal A \subset \cal P [n] be if it does not contain A,B with |A\setminus B| = 1? Our aim in this paper is to show that any such family has size at most \frac{2+o(1)}{n} \binom {n}{\lfloor n/2\rfloor }. This is tight…
An on-line chain partitioning algorithm receives a poset, one element at a time, and irrevocably assigns the element to one of the chains. Over 30 years ago, Szemer\'edi proved that any on-line algorithm could be forced to use…
An ordered $r$-uniform matching of size $n$ is a collection of $n$ pairwise disjoint $r$-subsets of a linearly ordered set of $rn$ vertices. For $n=2$, such a matching is called an $r$-pattern, as it represents one of $\tfrac12\binom{2r}r$…
A lattice of integers is the collection of all linear combinations of a set of vectors for which all entries of the vectors are integers and all coefficients in the linear combinations are also integers. Lattice reduction refers to the…
Let $\mathcal{P}(n)$ denote the power set of $[n]$, ordered by inclusion, and let $\mathcal{P}(n,p)$ be obtained from $\mathcal{P}(n)$ by selecting elements from $\mathcal{P}(n)$ independently at random with probability $p$. A classical…
Consider $S$, a set of $n$ points chosen uniformly at random and independently from the unit hypercube of dimension $t>2$. Order $S$ by using the Cartesian product of the $t$ standard orders of $[0,1]$. We determine a constant $\bar x(t)<e$…
Let $L(m,n)$ denote Young's lattice, consisting of all partitions whose Young diagrams are contained within an $m\times n$ rectangle. It is a classical result that the partially ordered set $L(m,n)$ is rank-symmetric, rank-unimodal, and…
We establish a lower bound of $2^n$ conditional branches for deciding the satisfiability of the conjunction of any two Boolean formulas from a set called a full representation of Boolean functions of $n$ variables - a set containing a…
Given a finite poset $\mathcal{P}$, a family $\mathcal{F}$ of elements in the Boolean lattice is induced-$\mathcal{P}$-saturated if $\mathcal{F}$ contains no copy of $\mathcal{P}$ as an induced subposet but every proper superset of…
In 1974, Erd\H{o}s and Kleitman conjectured that if a family $\mathcal{F}\subseteq 2^{[n]}$ contains no matching of size \(s\) and is maximal with respect to this property, then $ |\mathcal{F}|\ge \left(1-2^{-(s-1)}\right)\cdot 2^{n}. $ For…