Weak embeddings of posets to the Boolean lattice

The goal of this paper is to prove that several variants of deciding whether a poset can be (weakly) embedded into a small Boolean lattice, or to a few consecutive levels of a Boolean lattice, are NP-complete, answering a question of Griggs and of Patkos. As an equivalent reformulation of one of these problems, we also derive that it is NP-complete to decide whether a given graph can be embedded to the two middle levels of some hypercube

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)

Finitely generated free Heyting algebras via Birkhoff duality and coalgebra

Algebras axiomatized entirely by rank 1 axioms are algebras for a functor and thus the free algebras can be obtained by a direct limit process. Dually, the final coalgebras can be obtained by an inverse limit process. In order to explore the limits of this method we look at Heyting algebras which have mixed rank 0-1 axiomatizations. We will see that Heyting algebras are special in that they are almost rank 1 axiomatized and can be handled by a slight variant of the rank 1 coalgebraic methods

Incomparable Copies of a Poset in the Boolean Lattice

Let (Formula presented.) be the poset generated by the subsets of [n] with the inclusion relation and let (Formula presented.) be a finite poset. We want to embed (Formula presented.) into (Formula presented.) as many times as possible such that the subsets in different copies are incomparable. The maximum number of such embeddings is asymptotically determined for all finite posets (Formula presented.) as (Formula presented.), where (Formula presented.) denotes the minimal size of the convex hull of a copy of (Formula presented.). We discuss both weak and strong (induced) embeddings

Ramsey numbers for partially-ordered sets

We present a refinement of Ramsey numbers by considering graphs with a partial ordering on their vertices. This is a natural extension of the ordered Ramsey numbers. We formalize situations in which we can use arbitrary families of partially-ordered sets to form host graphs for Ramsey problems. We explore connections to well studied Tur\'an-type problems in partially-ordered sets, particularly those in the Boolean lattice. We find a strong difference between Ramsey numbers on the Boolean lattice and ordered Ramsey numbers when the partial ordering on the graphs have large antichains.Comment: 18 pages, 3 figures, 1 tabl

Contact semilattices

We devise exact conditions under which a join semilattice with a weak contact relation can be semilattice embedded into a Boolean algebra with an overlap contact relation, equivalently, into a distributive lattice with additive contact relation. A similar characterization is proved with respect to Boolean algebras and distributive lattices with weak contact, not necessarily additive, nor overlap.Comment: v3: noticed that former Condition (D2-) is pleonastic; added two new equivalent conditions in Theorem 3.2. We realized all this after the paper has been published: variations with respect to the published version are printed in a blue character. v2: solved a problem left open in v1; added a counterexample; a few fixe

On the Dual Canonical Monoids

We investigate the conjugacy decomposition, nilpotent variety, the Putcha monoid, as well as the two-sided weak order on the dual canonical monoids