Poset Ramsey number R(P,Qn)R(P,Q_n). III. Chain Compositions and Antichains

An induced subposet $(P_2,\le_2)$ of a poset $(P_1,\le_1)$ is a subset of $P_1$ such that for every two $X,Y\in P_2$, $X\le_2 Y$ if and only if $X\le_1 Y$. The Boolean lattice $Q_n$ of dimension $n$ is the poset consisting of all subsets of $\{1,\dots,n\}$ ordered by inclusion. Given two posets $P_1$ and $P_2$ the poset Ramsey number $R(P_1,P_2)$ is the smallest integer $N$ such that in any blue/red coloring of the elements of $Q_N$ there is either a monochromatically blue induced subposet isomorphic to $P_1$ or a monochromatically red induced subposet isomorphic to $P_2$. We provide upper bounds on $R(P,Q_n)$ for two classes of $P$: parallel compositions of chains, i.e.\ posets consisting of disjoint chains which are pairwise element-wise incomparable, as well as subdivided $Q_2$, which are posets obtained from two parallel chains by adding a common minimal and a common maximal element. This completes the determination of $R(P,Q_n)$ for posets $P$ with at most $4$ elements. If $P$ is an antichain $A_t$ on $t$ elements, we show that $R(A_t,Q_n)=n+3$ for $3\le t\le \log \log n$. Additionally, we briefly survey proof techniques in the poset Ramsey setting $P$ versus $Q_n$.Comment: 20 pages, 23 figures. Merged with arXiv:2205.0227

Poset Ramsey Number R(P, Qn). I. Complete Multipartite Posets

A poset $(P′,≤_{P′})$ contains a copy of some other poset $(P,≤_P)$ if there is an injection $f:P′→P$ where for every $X,Y∈P, X≤_PY$ if and only if $f(X)≤_{P′}f(Y)$. For any posets $P$ and $Q$, the poset Ramsey number $R(P, Q)$ is the smallest integer $N$ such that any blue/red coloring of a Boolean lattice of dimension $N$ contains either a copy of $P$ with all elements blue or a copy of $Q$ with all elements red. A complete ℓ-partite poset $K_{t1,…,tℓ}$ is a poset on $∑^ℓ_{i=1}t_i$ elements, which are partitioned into $ℓ$ pairwise disjoint sets $A^i$ with $|A^i|=t_i, 1≤i≤ℓ$, such that for any two $X∈A^i$ and $Y∈A^j$, $X<Y$ if and only if $i<j$. In this paper we show that $R(K_{t1,…,tℓ, }Q_n)≤n+\frac{(2+on(1))ℓn}{logn}$

Poset Ramsey number R(P,Qn)R(P,Q_n). III. N-shaped poset

Given partially ordered sets (posets) $(P, \leq_P)$ and $(P', \leq_{P'})$, we say that $P'$ contains a copy of $P$ if for some injective function $f\colon P\rightarrow P'$ and for any $A, B\in P$, $A\leq _P B$ if and only if $f(A)\leq_{P'} f(B)$. For any posets $P$ and $Q$, the poset Ramsey number $R(P,Q)$ is the least positive integer $N$ such that no matter how the elements of an $N$-dimensional Boolean lattice are colored in blue and red, there is either a copy of $P$ with all blue elements or a copy of $Q$ with all red elements. We focus on the poset Ramsey number $R(P, Q_n)$ for a fixed poset $P$ and an $n$-dimensional Boolean lattice $Q_n$, as $n$ grows large. It is known that $n+c_1(P) \leq R(P,Q_n) \leq c_2(P) n$, for positive constants $c_1$ and $c_2$. However, there is no poset $P$ known, for which $R(P, Q_n)> (1+\epsilon)n$, for $\epsilon >0$. This paper is devoted to a new method for finding upper bounds on $R(P, Q_n)$ using a duality between copies of $Q_n$ and sets of elements that cover them, referred to as blockers. We prove several properties of blockers and their direct relation to the Ramsey numbers. Using these properties we show that $R(\mathcal{N},Q_n)=n+\Theta(n/\log n)$, for a poset $\mathcal{N}$ with four elements $A, B, C,$ and $D$, such that $A<C$, $B<D$, $B<C$, and the remaining pairs of elements are incomparable.Comment: 19 pages, 6 figure

Generalizations of Eulerian partially ordered sets, flag numbers, and the Mobius function

A partially ordered set is r-thick if every nonempty open interval contains at least r elements. This paper studies the flag vectors of graded, r-thick posets and shows the smallest convex cone containing them is isomorphic to the cone of flag vectors of all graded posets. It also defines a k-analogue of the Mobius function and k-Eulerian posets, which are 2k-thick. Several characterizations of k-Eulerian posets are given. The generalized Dehn-Sommerville equations are proved for flag vectors of k-Eulerian posets. A new inequality is proved to be valid and sharp for rank 8 Eulerian posets

Classification of the factorial functions of Eulerian binomial and Sheffer posets

We give a complete classification of the factorial functions of Eulerian binomial posets. The factorial function B(n) either coincides with $n!$, the factorial function of the infinite Boolean algebra, or $2^{n-1}$, the factorial function of the infinite butterfly poset. We also classify the factorial functions for Eulerian Sheffer posets. An Eulerian Sheffer poset with binomial factorial function $B(n) = n!$ has Sheffer factorial function D(n) identical to that of the infinite Boolean algebra, the infinite Boolean algebra with two new coatoms inserted, or the infinite cubical poset. Moreover, we are able to classify the Sheffer factorial functions of Eulerian Sheffer posets with binomial factorial function $B(n) = 2^{n-1}$ as the doubling of an upside down tree with ranks 1 and 2 modified. When we impose the further condition that a given Eulerian binomial or Eulerian Sheffer poset is a lattice, this forces the poset to be the infinite Boolean algebra $B_X$ or the infinite cubical lattice $C_X^{< \infty}$. We also include several poset constructions that have the same factorial functions as the infinite cubical poset, demonstrating that classifying Eulerian Sheffer posets is a difficult problem.Comment: 23 pages. Minor revisions throughout. Most noticeable is title change. To appear in JCT

On the M\"obius Function and Topology of General Pattern Posets

We introduce a formal definition of a pattern poset which encompasses several previously studied posets in the literature. Using this definition we present some general results on the M\"obius function and topology of such pattern posets. We prove our results using a poset fibration based on the embeddings of the poset, where embeddings are representations of occurrences. We show that the M\"obius function of these posets is intrinsically linked to the number of embeddings, and in particular to so called normal embeddings. We present results on when topological properties such as Cohen-Macaulayness and shellability are preserved by this fibration. Furthermore, we apply these results to some pattern posets and derive alternative proofs of existing results, such as Bj\"orner's results on subword order.Comment: 28 Page

Operads with compatible CL-shellable partition posets admit a Poincar\'e-Birkhoff-Witt basis

In 2007, Vallette built a bridge across posets and operads by proving that an operad is Koszul if and only if the associated partition posets are Cohen-Macaulay. Both notions of being Koszul and being Cohen-Macaulay admit different refinements: our goal here is to link two of these refinements. We more precisely prove that any (basic-set) operad whose associated posets admit isomorphism-compatible CL-shellings admits a Poincar\'e-Birkhoff-Witt basis. Furthermore, we give counter-examples to the converse