Related papers: A sharp threshold for a random version of Sperner'…

Sperner's bound on the size of an antichain in the lattice P(S) of subsets of a finite set S has been generalized in three different directions: by Erdos to subsets of P(S) in which chains contain at most r elements; by Meshalkin to certain…

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…

In the Boolean lattice, Sperner's, Erd\H{o}s'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…

For an odd integer $n=2d-1$, let $\mathcal{B}(n, d)$ be the subgraph of the hypercube $Q_n$ induced by the two largest layers. In this paper, we describe the typical structure of independent sets in $\mathcal{B}(n, d)$ and give precise…

In this paper we show that for any poset $P$ that is not an antichain, the number of induced $P$-free families in the Boolean lattice $2^{[n]}$ is at most $ 2^{O(\mathrm{La}^*(n,P))}$, where $\mathrm{La}^*(n,P)$ denotes the the largest size…

For every integer $n$ with $n \geq 6$, we prove that the Boolean dimension of a poset consisting of all the subsets of $\{1,\dots,n\}$ equipped with the inclusion relation is strictly less than $n$.

Let $F(n,k)$ ($f(n,k)$) denote the maximum possible size of the smallest color class in a (partial) $k$-coloring of the Boolean lattice $B_n$ that does not admit a rainbow antichain of size $k$. The value of $F(n,3)$ and $f(n,2)$ has been…

We investigate the structure of the Medvedev lattice as a partial order. We prove that every interval in the lattice is either finite, in which case it is isomorphic to a finite Boolean algebra, or contains an antichain of size…

Given a set $X$, a collection $\mathcal{F} \subset \mathcal{P}(X)$ is said to be $k$-Sperner if it does not contain a chain of length $k+1$ under set inclusion and it is saturated if it is maximal with respect to this probability. Gerbner…

We prove a "supersaturation-type" extension of both Sperner's Theorem (1928) and its generalization by Erdos (1945) to k-chains. Our result implies that a largest family whose size is x more than the size of a largest k-chain free family…

We consider 'supersaturation' problems in partially ordered sets (posets) of the following form. Given a finite poset $P$ and an integer $m$ greater than the cardinality of the largest antichain in $P$, what is the minimum number of…

Let $P$ be a partially ordered set. If the Boolean lattice $(2^{[n]},\subset)$ can be partitioned into copies of $P$ for some positive integer $n$, then $P$ must satisfy the following two trivial conditions: (1) the size of $P$ is a power…

The width of a poset is the size of its largest antichain. Sperner's theorem states that $(2^{[n]},\subset)$ is a poset whose width equals the size of its largest layer. We show that Hamming ball posets also have this property. This extends…

Let $\mathcal{G}$ be the set of all connected graphs on vertex set $[n]$. Define the partial ordering $<$ on $\mathcal{G}$ as follows: for $G,H\in \mathcal{G}$ let $G<H$ if $E(G)\subset E(H)$. The poset $(\mathcal{G},<)$ is graded, each…

A central theorem in combinatorics is Sperner's Theorem, which determines the maximum size of a family $\mathcal{F}\subseteq \mathcal{P}(n)$ that does not contain a $2$-chain $F_1\subsetneq F_2$. Erd\H{o}s later extended this result and…

For a partially ordered set P, we denote by Co(P) the lattice of order-convex subsets of P. We find three new lattice identities, (S), (U), and (B), such that the following result holds. Theorem. Let L be a lattice. Then L embeds into some…

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…

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 $y,z<w$.…

We develop a powerful tool for embedding any tree poset $P$ of height $k$ in the Boolean lattice which allows us to solve several open problems in the area. We show that: * If $H$ is a family in $B_n$ with $|H|\ge (q-1+\varepsilon){n\choose…

We prove that the largest $Q_2$-free family of subsets of $[n]$ which contains sets of at most three different sizes has at most $(3 + 2\sqrt {3})N/3 + o(N) \approx 2.1547N + o(N)$ members, where $N = {n \choose {\lfloor n/2 \rfloor}}$.…