Generalizations of Sperner\u27s Theorem: Packing Posets, Families Forbidding Posets, and Supersaturation

Sperner\u27s Theorem is a well known theorem in extremal set theory that gives the size of the largest antichain in the poset that is the Boolean lattice. This is equivalent to finding the largest family of subsets of an $n$-set, $[n]:=\{1,2,\dots,n\}$, such that the family is constructed from pairwise unrelated copies of the single element poset. For a poset $P$, we are interested in maximizing the size of a family $\mathcal{F}$ of subsets of $[n]$, where each maximally connected component of $\mathcal{F}$ is a copy of $P$, and finding the extreme configurations that achieve this value. For instance, Sperner showed that when $P$ is one element, $\dbinom{n}{\lfloor \frac{n}{2}\rfloor}$ is the maximum number of copies of $P$ and that this is only achieved by taking subsets of a middle size. Griggs, Stahl, and Trotter have shown that when $P$ is a chain on $k$ elements, $\dfrac{1}{2^{k-1}}\dbinom{n}{\lfloor \frac{n}{2}\rfloor}$ is asymptotically the maximum number of copies of $P$. We find the extreme families for a packing of chains, answering a conjecture of Griggs, Stahl, and Trotter, as well as finding the extreme packings of certain other posets. For the general poset $P$, we prove that the maximum number of unrelated copies of $P$ is asymptotic to a constant times $\dbinom{n}{\lfloor \frac{n}{2}\rfloor}$. Moreover, the constant has the form $\dfrac{1}{c(P)}$, where $c(P)$ is the size of the smallest convex closure over all embeddings of $P$ into the Boolean lattice. Sperner\u27s Theorem has been generalized by looking for $\operatorname{La}(n,P)$, the size of a largest family of subsets of an $n$-set that does not contain a general poset $P$ in the family. We look at this generalization, exploring different techniques for finding an upper bound on $\operatorname{La}(n,P)$, where $P$ is the diamond. We also find all the families that achieve $\operatorname{La}(n,\{\mathcal{V},\Lambda\})$, the size of the largest family of subsets that do not contain either of the posets $\mathcal{V}$ or $\Lambda$. We also consider another generalization of Sperner\u27s theorem, supersaturation, where we find how many copies of $P$ are in a family of a fixed size larger than $\operatorname{La}(n,P)$. We seek families of subsets of an $n$-set of given size that contain the fewest $k$-chains. Erd\H{o}s 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)

Structure determination of disordered materials from diffraction data

We show that the information gained in spectroscopic experiments regarding the number and distribution of atomic environments can be used as a valuable constraint in the refinement of the atomic-scale structures of nanostructured or amorphous materials from pair distribution function (PDF) data. We illustrate the effectiveness of this approach for three paradigmatic disordered systems: molecular C60, a-Si, and a-SiO2 . Much improved atomistic models are attained in each case without any a-priori assumptions regarding coordination number or local geometry. We propose that this approach may form the basis for a generalised methodology for structure "solution" from PDF data applicable to network, nanostructured and molecular systems alike.Comment: 4 pages, 3 figures, set out as for PR

Highly polarized alkenes as organocatalysts for the polymerization of lactones and trimethylene carbonate

In this work, the activity of N-heterocyclic olefins (NHOs), a newly emerging class of organopolymerization catalyst, is investigated to affect the metal-free polymerization of lactones and trimethylene carbonate (TMC). A decisive structure−activity relationship is revealed. While catalysts of the simplest type bearing an exocyclic CH2 moiety polymerize L-lactide (L-LA) and δ-valerolactone (δ-VL) in a non-living and non-quantitative manner, the introduction of methyl substituents on the exocyclic carbon radically changes this behavior. 2-Isopropylidene-1,3,4,5-tetramethylimidazoline is found to be highly active for a range of monomers such as L-LA, δ-VL, ε-caprolactone (ε-CL), and TMC, with quantitative conversion occurring within seconds with catalyst loadings of just 0.2 mol %. The high activity of this NHO further enables the ring-opening polymerization (ROP) of the macrolactone ω-pentadecalactone (PDL). However, this broad applicability is offset by a lack of control over the polymerizations, including side reactions as a consequence of its strong basicity. To overcome this, a saturated, imidazolinium-derived analogue was synthesized and subsequently demonstrated to possess a harnessed reactivity which enables it to polymerize both L-LA and TMC in a controlled manner (ĐM < 1.2). NMR spectroscopic and MALDI-ToF MS experiments highlight the differences in polymerization pathways for 2-methylene-1,3,4,5-tetramethylimidazoline, in which the exocyclic carbon is not substituted, in contrast to 2-isopropylidene-1,3,4,5-tetramethylimidazoline, with the former operating via its nucleophilicity and the latter acting as a base with enolizable δ-VL

Selective and Sequential Catalytic Chemical Depolymerization and Upcycling of Mixed Plastics

Chemical recycling to monomer (CRM) provides a useful technique to allow for polymer-to-monomer-to-polymer circular economies. A significant challenge remains, however, in the treatment of mixed plastics by CRM in which unselective depolymerization requires either presorting of plastics or purification processes postdepolymerization, both of which add cost to waste plastic processing. We report a simple, yet selective, chemical depolymerization of three commonly used polymers, poly(lactic acid) (PLA), bisphenol A polycarbonate (BPA-PC), and polyethylene terephthalate (PET), using inexpensive and readily available common metal salt/organobase dual catalysts. By a judicious choice of catalyst and conditions, selective and sequential depolymerization of mixtures of the polymers was demonstrated. Furthermore, the potential for upcycling of polymers to value-added monomers was explored through the application of alternative nucleophiles within the depolymerization.</p