Related papers: Maximal antichains of subsets II: Constructions

This is the second of two papers investigating for which positive integers $m$ there exists a maximal antichain of size $m$ in the Boolean lattice $B_n$ (the power set of $[n]:=\{1,2,\dots,n\}$, ordered by inclusion). In the first part, the…

Extending a classical theorem of Sperner, we characterize the integers $m$ such that there exists a maximal antichain of size $m$ in the Boolean lattice $B_n$, that is, the power set of $[n]:=\{1,2,\dots,n\}$, ordered by inclusion. As an…

How large an antichain can we find inside a given downset in the lattice of subsets of [n]? Sperner's theorem asserts that the largest antichain in the whole lattice has size the binomial coefficient C(n, n/2); what happens for general…

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 $\mathcal{P}(n)$ consists of all subsets of $[n] = \{1,\dots, n\}$ partially ordered under the containment relation. Sperner's Theorem states that the largest antichain of the Boolean lattice is given by a middle layer:…

We show that, for $pn \to \infty$, the largest set in a $p$-random sub-family of the power set of $\{1, \ldots, n\}$ containing no $k$-chain has size $( k - 1 + o(1) ) p \binom{n}{n/2}$ with high probability. This confirms a conjecture of…

Let $n\geqslant 4$ be a natural number, and let $K$ be a set $K\subseteq [n]:={1,2,...,n}$. We study the problem to find the smallest possible size of a maximal family $\mathcal{A}$ of subsets of $[n]$ such that $\mathcal{A}$ contains only…

Building on classical theorems of Sperner and Kruskal-Katona, we investigate antichains $\mathcal F$ in the Boolean lattice $B_n$ of all subsets of $[n]:=\{1,2,\dots,n\}$, where $\mathcal F$ is flat, meaning that it contains sets of at most…

We give a short and self-contained argument that shows that, for any positive integers $t$ and $n$ with $t =O\Bigl(\frac{n}{\log n}\Bigr)$, the number $\alpha([t]^n)$ of antichains of the poset $[t]^n$ is at most…

We establish one-to-one correspondences between maximal antichains in products of two finite linear orders and other mathematical objects, such as certain alignments of two strings, walks on a grid, lattice paths, words of two or three…

We present two results on maximal antichains in the strict chain product poset $[t_1+1]\times[t_2+1]\times\ldots\times[t_n+1]$. First, we prove that these maximal antichains are also maximum. Second, we prove that there is a bijection…

We present new exact and asymptotic results about the size of the largest antichain in the product of $n$ linear orders.

Answering several questions of Duffus, Frankl and R\"odl, we give asymptotics for the logarithms of (i) the number of maximal antichains in the n-dimensional Boolean algebra and (ii) the numbers of maximal independent sets in the covering…

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…

One of the most classical results in extremal set theory is Sperner's theorem, which says that the largest antichain in the Boolean lattice $2^{[n]}$ has size $\Theta\big(\frac{2^n}{\sqrt{n}}\big)$. Motivated by an old problem of Erd\H{o}s…

Let $[n] = \{1, 2, \ldots, n\}$ and let $2^{[n]}$ be the collection of all subsets of $[n]$ ordered by inclusion. ${\cal C} \subseteq 2^{[n]}$ is a {\em cutset} if it meets every maximal chain in $2^{[n]}$, and the {\em width} of ${\cal C}…

The Boolean lattice $2^{[n]}$ is the family of all subsets of $[n]=\{1,\dots,n\}$ ordered by inclusion, and a chain is a family of pairwise comparable elements of $2^{[n]}$. Let $s=2^{n}/\binom{n}{\lfloor n/2\rfloor}$, which is the average…

We provide precise asymptotics for the number of antichains in the poset $\{0,1,2\}^n$, answering a question of Sapozhenko. Finding improved estimates for this number was also a problem suggested by Noel, Scott, and Sudakov, who obtained…

We study maximum antichains in two posets related to quiver representations. Firstly, we consider the set of isomorphism classes of indecomposable representations ordered by inclusion. For various orientations of the Dynkin diagram of type…

Let $(\mathcal{P},\leqslant)$ be a finite poset. Define the numbers $a_1,a_2,\ldots$ (respectively, $c_1,c_2,\ldots$) so that $a_1+\ldots+a_k$ (respectively, $c_1+\ldots+c_k$) is the maximal number of elements of $\mathcal{P}$ which may be…