Tree posets: Supersaturation, enumeration, and randomness

We study the problem of deciding, whether a given partial order is embeddable into two consecutive layers of a Boolean lattice. Employing an equivalent condition for such em- beddability similar to the one given by J. Mittas and K. Reuter [5], we prove that the decision pro

Given a graded poset we say a family is centered if it is obtained by ‘taking sets as close to the middle layer as possible.’ A poset is said to have the centeredness property if for any , among all families of size in , centered families contain the minimum number of comparable pairs. Kleitman showed that the Boolean lattice has the centeredness property. It was conjectured by Noel, Scott, and Sudakov, and by Balogh and Wagner, that the poset also has the centeredness property, provided is sufficiently large compared with . We show that this conjecture is false for all and investigate the range of for which it holds. Further, we improve a result of Noel, Scott, and Sudakov by showing that the poset of subspaces of has the centeredness property. Several open questions are also given.

The largest projective cube-free subsets of [formula omitted

Active student engagement is key to this classroom-tested combinatorics text, boasting 1200+ carefully designed problems, ten mini-projects, section warm-up problems, and chapter opening problems. The author – an award-winning teacher – writes in a conversational style, keeping the reader in mind on every page. Students will stay motivated through glimpses into current research trends and open problems as well as the history and global origins of the subject. All essential topics are covered, including Ramsey theory, enumerative combinatorics including Stirling numbers, partitions of integers, the inclusion-exclusion principle, generating functions, introductory graph theory, and partially ordered sets. Some significant results are presented as sets of guided problems, leading readers to discover them on their own. More than 140 problems have complete solutions and over 250 have hints in the back, making this book ideal for self-study. Ideal for a one semester upper undergraduate course, prerequisites include the calculus sequence and familiarity with proofs.

We apply the graph container method to prove a number of counting results for the Boolean lattice . In particular, we: Give a partial answer to a question of Sapozhenko estimating the number of t error correcting codes in , and we also give an upper bound on the number of transportation codes; Provide an alternative proof of Kleitman's theorem on the number of antichains in and give a two‐coloured analogue; Give an asymptotic formula for the number of (p, q)‐tilted Sperner families in ; Prove a random version of Katona's t‐intersection theorem. In each case, to apply the container method, we first prove corresponding supersaturation results. We also give a construction which disproves two conjectures of Ilinca and Kahn on maximal independent sets and antichains in the Boolean lattice. A number of open questions are also given. © 2016 Wiley Periodicals, Inc. Random Struct. Alg., 49, 845–872, 2016

Four distinct elements a, b, c, and d of a poset form a diamond if \(a< b<d\) and \(a<c<d\). A subset of a poset is diamond-free if no four elements of the subset form a diamond. Even in the Boolean lattices, finding the size of the largest diamond-free subset remains an open problem. In this paper, we consider the linear lattices—poset of subspaces of a finite dimensional vector space over a finite field of order q—and extend the results of Griggs et al. (J. Combin. Theory Ser. A 119(2):310–322, 2012) on the Boolean lattices, to prove that the number of elements of a diamond-free subset of a linear lattice can be no larger than \(2+\frac {1}{q+1}\) times the width of the lattice, so that this fraction tends to 2 as \(q \longrightarrow \infty \). In addition, using an algebraic technique, we introduce so-called diamond matchings, and prove that for linear lattices of dimensions up to 5, the size of a largest diamond-free subset is equal to the sum of the largest two rank numbers of the lattice.

ABSTRACTThe Boolean lattice consists of all subsets of partially ordered under the containment relation. Sperner's Theorem states that the largest antichain of the Boolean lattice is given by a middle layer: the collection of all sets of size , or also, if is odd, the collection of all sets of size . Given , choose each subset of with probability independently. We show that for every constant , the largest antichain among these subsets is also given by a middle layer, with probability tending to 1 as tends to infinity. This is best possible, and we also characterize the largest antichains for every constant . Our proof is based on some new variations of Sapozhenko's graph container method.

We consider the problem of invariance of distillable entanglement D and quantum capacities Q under erasure of information about single copy of quantum state or channel respectively. We argue that any 2 ⊗N two-way distillable state is still two-way distillable after erasure of single copy information. For some known distillation protocols the obtained two-way distillation rate is the same as if Alice and Bob knew the state from the very beginning. The isomorphism between quantum states and quantum channels is also investigated. In particular it is pointed out that any transmission rate down the channel is equal to distillation rate with formal LOCC-like superoperator that uses in general nonphysical Alice actions. This allows to we prove that if given channel Λ has nonzero capacity (Q → or Q ⟺) then the corresponding quantum state ϱ(Λ) has nonzero distillable entanglement (D → or D ⟺). Follwoing the latter arguments are provided that any channel mapping single qubit into N level system allows for reliable two-way transmission after erasure of information about single copy. Some open problems are discussed.

We show that symmetric Venn diagrams for $n$ sets exist for every prime $n$, settling an open question. Until this time, $n=11$ was the largest prime for which the existence of such diagrams had been proven, a result of Peter Hamburger. We show that the problem can be reduced to finding a symmetric chain decomposition, satisfying a certain cover property, in a subposet of the Boolean lattice ${\cal B}_n$, and prove that such decompositions exist for all prime $n$. A consequence of the approach is a constructive proof that the quotient poset of ${\cal B}_n$, under the relation "equivalence under rotation", has a symmetric chain decomposition whenever $n$ is prime. We also show how symmetric chain decompositions can be used to construct, for all $n$, monotone Venn diagrams with the minimum number of vertices, giving a simpler existence proof.

Many researchers have had fun searching for and rendering symmetric Venn diagrams, culminating in the recent result of Griggs, Killian, and Savage ("Venn Diagrams and Symmetric Chain Decompositions in the Boolean Lattice," Electronic Journal of Combinatorics, 11 (2004), R2), that symmetric Venn diagrams on n curves exist if and only if n is prime. Our purpose here is to point out ways to prolong the fun by introducing and finding the basic properties of Venn and near-Venn diagrams that satisfy relaxed notions of symmetry, while leaving tantalizing open problems. A symmetric Venn diagram is one that possesses a rotational symmetry in the plane. A monochrome symmetric Venn diagram is one that is rotationally symmetric when the colours of the curves are ignored. A necessary condition for the existence of a monochrome symmetric Venn diagram is that the number of curves be a prime power. We specify conditions under which all curves must be non-congruent and give examples of small visually striking monochrome symmetric Venn diagrams found by algorithmic searches. A Venn diagram partitions the plane into 2n open regions. For non-prime n we also consider symmetric diagrams where the number of regions is as close to 2n as possible, both larger and smaller.

Supersaturation in posets and applications involving the container method

An upper bound on the size of diamond-free families of sets

Packing and covering [formula omitted]-chain free subsets in Boolean lattices

In this article, we determine the probability of existence of small lattices in random subsets of a Boolean lattice. Furthermore, we address some Ramsey- and Turán-type questions. Analogous questions have been studied extensively for random graphs, but it turns out that the situation for Boolean lattices is quite different. © 1998 John Wiley & Sons, Inc. Random Struct. Alg., 13: 383–407, 1998

The integration of smart materials with metamaterials offers significant research value in additive manufacturing. This paper proposes a novel approach combining shape memory polymers (SMP) with zero Poisson’s ratio (ZPR) metamaterials to create better space-filling, lightweight, and high-energy absorbing materials. A cellular structure with three deformation processes was designed by adjusting the rib arm lengths of the ZPR lattice AuxHex, forming a 3D gradient structure. The cellular units were also prepared using the dual-nozzle fusion filament printing technique, and the energy absorption and shape recovery properties of various hierarchical gradient structures, compared to the uniform structure, were simulated and analyzed using the viscoelastic intrinsic model. The results indicate that all gradient structures exhibit superior energy absorption capabilities compared to uniform materials, with the energy absorption performance of the optimal gradient material being 44.32% higher than that of the nongradient material, and they also enhance the shape recovery performance to some extent. Furthermore, it was observed that when a lattice layer, which avoids internal rib contact throughout the compression process, is positioned in the middle layer, it harmonizes the deformation of the upper and lower layers, thereby improving overall stability. This configuration enhances both energy absorption and partial shape recovery performance, offering a novel approach for the design of smart material assemblies with graded metamaterial structures. This study provides a new perspective and methodology for the future design of supergradient smart material collections.