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,… ,n} , ordered by inclusion). In the first part, the sizes of maximal antichains have been characterized. Here we provide an alternative construction with the benefit of showing that almost all sizes of maximal antichains can be obtained using antichains containing only l-sets and (l+1) -sets for some l.

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 Bn, that is, the power set of [n]:={1,2,… ,n} , ordered by inclusion. As an important ingredient in the proof, we initiate the study of an extension of the Kruskal-Katona theorem which is of independent interest. For given positive integers t and k, we ask which integers s have the property that there exists a family ℱ of k-sets with |ℱ|=t such that the shadow of ℱ has size s, where the shadow of ℱ is the collection of (k − 1)-sets that are contained in at least one member of ℱ . We provide a complete answer for t⩽ k+1 . Moreover, we prove that the largest integer which is not the shadow size of any family of k-sets is √(2)k^3/2+√(8)k^5/4+O(k) .

Let $L(G)$ denote the maximum number of leaves in any spanning tree of a connected graph $G$. We show the (known) result that for the $n$-cube $Q_n$, $L(Q_n) \sim 2^n = |V(Q_n)|$ as $n\rightarrow \infty$. Examining this more carefully, consider the minimum size of a connected dominating set of vertices $\gamma_c(Q_n)$, which is $2^n-L(Q_n)$ for $n\ge2$. We show that $\gamma_c(Q_n)\sim 2^n/n$. We use Hamming codes and an "expansion" method to construct leafy spanning trees in $Q_n$.

This is the second in a sequence of three papers investigating the question 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 previous paper we characterized those $m$ between $\binom{n}{\lceil n/2\rceil}-\lceil n/2\rceil^2$ and the maximum size $\binom{n}{\lceil n/2 \rceil}$ that are not sizes of maximal antichains. In this paper we show that all smaller $m$ are sizes of maximal antichains.

Building on classical theorems of Sperner and Kruskal-Katona, we investigate antichains $\mathcal {F}$ in the Boolean lattice Bn of all subsets of $[n]:=\{1,2,\dots ,n\}$ , where $\mathcal {F}$ is flat, meaning that it contains sets of at most two consecutive sizes, say $\mathcal {F}=\mathcal {A}\cup {\mathscr{B}}$ , where $\mathcal {A}$ contains only k-subsets, while ${\mathscr{B}}$ contains only (k − 1)-subsets. Moreover, we assume $\mathcal {A}$ consists of the first m k-subsets in squashed (colexicographic) order, while ${\mathscr{B}}$ consists of all (k − 1)-subsets not contained in the subsets in $\mathcal {A}$ . Given reals α, β > 0, we say the weight of $\mathcal {F}$ is $\alpha \cdot |\mathcal {A}|+\beta \cdot |{\mathscr{B}}|$ . We characterize the minimum weight antichains $\mathcal {F}$ for any given n,k,α,β, and we do the same when in addition $\mathcal {F}$ is a maximal antichain. We can then derive asymptotic results on both the minimum size and the minimum Lubell function.

Let L(G) denote the maximum number of leaves in any spanning tree of a connected graph G. We show the (known) result that for the n-cube Qn, L(Qn) ∼ 2 = |V (Qn)| as n → ∞. Examining this more carefully, consider the minimum size of a connected dominating set of vertices γc(Qn), which is 2 n − L(Qn) for n ≥ 2. We show that γc(Qn) ∼ 2/n, which rather surprisingly is no larger than the asymptotic behavior of the domination number γ(Qn). We use Hamming codes and an “expansion” method to construct leafy spanning trees in Qn.

Extending a classical theorem of Sperner, we investigate the question for which positive integers m there exists a maximal antichain of sizem in the Boolean lattice Bn, that is, the power set of [n] := {1, 2, . . . , n}, ordered by inclusion. We characterize all such integers m in the range ( n ⌈n/2⌉ ) −⌈n/2⌉2 6 m 6 ( n ⌈n/2⌉ ) . As an important ingredient in the proof, we initiate the study of an extension of the Kruskal-Katona theorem which is of independent interest. For given positive integers t and k, we ask which integers s have the property that there exists a family F of k-sets with |F| = t such that the shadow of F has size s, where the shadow of F is the collection of (k − 1)-sets that are contained in at least one member of F . We provide a complete answer for the case t 6 k + 1. Moreover, we prove that the largest integer which is not the shadow size of any family of k-sets is √ 2k3/2 + 4 √ 8k5/4 +O(k).

Recently, Balogh--Morris--Samotij and Saxton--Thomason proved that hypergraphs satisfying some natural conditions have only few independent sets. Their main results already have several applications. However, the methods of proving these theorems are even more far reaching. The general idea is to describe some family of events, whose cardinality a priori could be large, only with a few certificates. Here, we show some applications of the methods, including counting $C_4$-free graphs, considering the size of a maximum $C_4$-free subgraph of a random graph and counting metric spaces with a given number of points. Additionally, we discuss some connections with the Szemer\'edi Regularity Lemma.

Increasing attention is being paid to the study of families of subsets of an n-set that contain no subposet P. Especially, we are interested in such families of maximum size given P and n. For certain P this problem is solved for general n, while for other P it is extremely challenging to find even an approximate solution for large n. It is conjectured that for any P, the maximum size is asymptotic to a constant times ([n/2]) where the constant is a certain integer depending on P. This survey has two purposes. First, we want to bring this exciting line of research to the attention of a wider audience. Second, we want to make experts aware of the broad range of recent progress in the area.

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 largest binomial coefficient. Moreover, the constant has the form $\frac{1}{c(P)}$, where $c(P)$ is the size of the smallest convex closure over all embeddings of $P$ into the Boolean lattice.

Given a finite poset P, we consider the largest size La(n,P) of a family F of subsets of [n]:={1,…,n} that contains no subposet P. This continues the study of the asymptotic growth of La(n,P); it has been conjectured that for all P, π(P):=limn→∞⁡La(n,P)/(n⌊n2⌋) exists and equals a certain integer, e(P). This is known to be true for paths, and for several more general families of posets, while for the simple diamond poset D2, even the existence of π frustratingly remains open. Here we develop theory to show that π(P) exists and equals the conjectured value e(P) for many new posets P. We introduce a hierarchy of properties for posets, each of which implies π=e, and some implying more precise information about La(n,P). The properties relate to the Lubell function of a family F of subsets, which is the average number of times a random full chain meets F. We present an array of examples and constructions that possess the properties.

We seek families of subsets of an n-set of given size that contain the fewest k-chains. We prove a “supersaturation-type” extension of both Sperner’s Theorem (1928) and its generalization by Erdős (1945). Erdős showed that a largest k-chain free family in the Boolean lattice is formed by taking all subsets of the (k 1) middle sizes. Our result implies that by taking this family together with x subsets of the k-th middle size, we obtain a family with the minimum number of k-chains, over all families of this size. We prove our result using the symmetric chain decomposition method of de Bruijn, van Ebbenhorst Tengbergen, and Kruyswijk (1951).

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 and that contains the minimum number of k-chains is the family formed by taking the middle (k-1) rows of the Boolean lattice and x elements from the k-th middle row. We prove our result using the symmetric chain decomposition method of de Bruijn, van Ebbenhorst Tengbergen, and Kruyswijk (1951).

Given a finite poset P, let \({\rm La}(n,P)\) denote the largest size of a family of subsets of an n-set that does not contain P as a (weak) subposet. We employ a combinatorial method, using partitions of the collection of all full chains of subsets of the n-set, to give simpler new proofs of the known asymptotic behavior of \({\rm La}(n,P)\), as n→∞, when P is the r-fork \(\mathcal {V}_{r}\), the four-element N poset \(\mathcal {N}\), and the four-element butterfly-poset \(\mathcal {B}\).

any n;k and positive real numbers ; , we determine all SFFA and all SMFA of minimum weight jAj + jBj . Based on this, asymptotic results on SMFA with minimum size and minimum BLYM value, respectively, are derived.

Given a finite poset P. we consider the largest size La(n, P) of a family of subsets of [n] := {1, ..., n} that contains no (weak) subposet P. This problem has been studied intensively in recent years, and it is conjectured that pi (P) := lim(n ->infinity)La(n, p)/((n)([n/2])) exists for general posets P, and, moreover, it is an integer. For k >= 2 let D(k) denote the k-diamond poset {A < B(1),..., B(k) < C}. We study the average number of times a random full chain meets a P-free family, called the Lubell function, and use it for P = D(k) to determine pi(D(k)) for infinitely many values k. A stubborn open problem is to show that m(D(2)) = 2; here we make progress by proving pi (D(2)) <= 2 (3/11) (if it exists). (C) 2011 Elsevier Inc. All rights reserved.

Given a finite poset P , we consider the largest size La ( n , P ) of a family of subsets of [ n ] := { 1 ,..., n } that contains no (weak) subposet P . This problem has been studied intensively in recent years, and it is conjectured that π ( P ) := lim n →∞

Graph labellings form an important graph theory model for the channel assignment problem. An optimum labelling usually depends on one or more parameters that ensure minimum separations between frequencies assigned to nearby transmitters. The study of spans and of the structure of optimum labellings as functions of such parameters has attracted substantial attention from researchers, leading to the introduction of real number graph labellings and λ-graphs. We survey recent results obtained in this area.

Let ⊂ 2 [ n ] be a family of subsets of {1, 2,. . ., n }. For any poset H , we say is H -free if does not contain any subposet isomorphic to H . Katona and others have investigated the behaviour of La( n , H ), which denotes the maximum size of H -free families ⊂ 2 [ n ] . Here we use a new approach, which is to apply methods from extremal graph theory and probability theory to identify new classes of posets H , for which La( n , H ) can be determined asymptotically as n → ∞ for various posets H , including two-end-forks, up-down trees, and cycles C 4 k on two levels.