Related papers: Boolean dimension of a Boolean lattice

For every integer $n$ with $n \geq 4$, we prove that the local dimension of a poset consisting of all the subsets of $\{1,\dots,n\}$ equipped with the inclusion relation is strictly less than $n$, answering a question of Kim, Martin,…

Let $P$ be a poset of size $2^k$ that has a greatest and a least element. We prove that, for sufficiently large $n$, the Boolean lattice $2^{[n]}$ can be partitioned into copies of $P$. This resolves a conjecture of Lonc.

We prove upper and lower bounds on the local dimension of any pair of layers of the Boolean lattice, and show that the local dimension of the first and middle layers of the $n$-dimensional Boolean lattice is asymptotically $\frac{n}{\log_2…

The dimension is a key measure of complexity of partially ordered sets. Small dimension allows succinct encoding. Indeed if $P$ has dimension $d$, then to know whether $x \leq y$ in $P$ it is enough to check whether $x\leq y$ in each of the…

Let $P$ be a partially ordered set. We prove that if $n$ is sufficiently large, then there exists a packing $\mathcal{P}$ of copies of $P$ in the Boolean lattice $(2^{[n]},\subset)$ that covers almost every element of $2^{[n]}$:…

Let $B_n$ be the poset generated by the subsets of $[n]$ with the inclusion as relation and let $P$ be a finite poset. We want to embed $P$ into $B_n$ as many times as possible such that the subsets in different copies are incomparable. The…

Dimension is a standard and well-studied measure of complexity of posets. Recent research has provided many new upper bounds on the dimension for various structurally restricted classes of posets. Bounded dimension gives a succinct…

For integers $0 \leq m \leq l \leq n-m$, the truncated Boolean lattice ${\cal B}_n(m,l)$ is the poset of all subsets of $[n] = \{1, 2, \ldots, n\}$ which have size at least $m$ and at most $l$. ${\cal C} \subseteq {\cal B}_n(m,l)$ is a {\em…

The Boolean lattice of dimension two, also known as the diamond, consists of four distinct elements with the following property: $A\subset B,C\subset D$. A diamond-free family in the $n$-dimensional Boolean lattice is a subposet such that…

The Boolean lattice $2^{[n]}$ is the power set of $[n]$ ordered by inclusion. A chain $c_{0}\subset...\subset c_{k}$ in $2^{[n]}$ is rank-symmetric, if $|c_{i}|+|c_{k-i}|=n$ for $i=0,...,k$; and it is symmetric, if $|c_{i}|=(n-k)/2+i$. We…

The $\mathcal{N}$ poset consists of four distinct sets $W,X,Y,Z$ such that $W\subset X$, $Y\subset X$, and $Y\subset Z$ where $W$ is not necessarily a subset of $Z$. A family $\mathcal{F}$ as a subposet of the $n$-dimensional Boolean…

We show that there is a constant $C>0$ such that for each integer $n\geq 1$, there is a poset on at most $2^{2n/3+C\sqrt{n}}$ elements that contains each $n$-element poset as an (induced) subposet.

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…

We are interested in maximizing the number of pairwise unrelated copies of a poset $P$ in the family of all subsets of $[n]$. We prove that for any $P$ the maximum number of unrelated copies of $P$ is asymptotic to a constant times the…

In 1989, Ne\v{s}et\v{r}il and Pudl\'ak posed the following challenging question: Do planar posets have bounded Boolean dimension? We show that every poset with a planar cover graph and a unique minimal element has Boolean dimension at most…

We explore lattice structures on integer binary relations (i.e. binary relations on the set $\{1, 2, \dots, n\}$ for a fixed integer $n$) and on integer posets (i.e. partial orders on the set $\{1, 2, \dots, n\}$ for a fixed integer $n$).…

A subposet $Q'$ of a poset $Q$ is a copy of a poset $P$ if there is a bijection $f$ between elements of $P$ and $Q'$ such that $x\leq y$ in $P$ iff $f(x)\leq f(y)$ in $Q'$. For posets $P, P'$, let the poset Ramsey number $R(P,P')$ be the…

We prove that there is a lattice embedded from every countable distributive lattice into the Boolean algebra of computable subsets of $\mathbb{N}$. Along the way, we discuss all relevant results about lattices, Boolean algebras and…

Planar posets can have arbitrarily large dimension. However, a planar poset of height $h$ has dimension at most $192h+96$, while a planar poset with $t$ minimal elements has dimension at most $2t+1$. In particular, a planar poset with a…

Let $La(n,P)$ be the maximum size of a family of subsets of $[n]=\{1,2,...,n\}$ not containing $P$ as a (weak) subposet. The diamond poset, denoted $B_{2}$, is defined on four elements $x,y,z,w$ with the relations $x<y,z$ and $y,z<w$.…