Forbidden subposet problems in the grid

For posets $P$ and $Q$, extremal and saturation problems about weak and strong $P$-free subposets of $Q$ have been studied mostly in the case $Q$ is the Boolean poset $Q_n$, the poset of all subsets of an $n$-element set ordered by inclusion. In this paper, we study some instances of the problem with $Q$ being the grid, and its connections to the Boolean case and to the forbidden submatrix problem

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

Forbidden subposet problems in the grid

For posets P and Q, extremal and saturation problems about weak and strong P-free subposets of Q have been studied mostly in the case Q is the Boolean poset Qn, the poset of all subsets of an n-element set ordered by inclusion. In this paper, we study some instances of the problem with Q being the grid, and its connections to the Boolean case and to the forbidden submatrix problem

Combinatorics of Topological Posets:\ Homotopy complementation formulas

We show that the well known {\em homotopy complementation formula} of Bj\"orner and Walker admits several closely related generalizations on different classes of topological posets (lattices). The utility of this technique is demonstrated on some classes of topological posets including the Grassmannian and configuration posets, $\widetilde{\mathbf G}_n(R)$ and $\exp_n(X)$ which were introduced and studied by V.~Vassiliev. Among other applications we present a reasonably complete description, in terms of more standard spaces, of homology types of configuration posets $\exp_n(S^m)$ which leads to a negative answer to a question of Vassilev raised at the workshop ``Geometric Combinatorics'' (MSRI, February 1997)

Problems in extremal graphs and poset theory

In this dissertation, we present three different research topics and results regarding such topics. We introduce partially ordered sets (posets) and study two types of problems concerning them-- forbidden subposet problems and induced-poset-saturation problems. We conclude by presenting results obtained from studying vertex-identifying codes in graphs. In studying forbidden subposet problems, we are interested in estimating the maximum size of a family of subsets of the $n$-set avoiding a given subposet. We provide a lower bound for the size of the largest family avoiding the $\mathcal{N}$ poset, which makes use of error-correcting codes. We also provide and upper and lower bound results for a $k$-uniform hypergraph that avoids a \emph{triangle}. Ferrara et al. introduced the concept of studying the minimum size of a family of subsets of the $n$-set avoiding an induced poset, called induced-poset-saturation. In particular, the authors provided a lower bound for the size of an induced-antichain poset and we improve on their lower bound result. Let $G=(V,E)$ be a graph with vertex set $V$ and edge set $E$. For any nonnegative integer $r$, let $B_r(v)$ denote the ball of radius $r$ around vertex $v\in V$. For a finite graph $G$, an $r$-vertex-identifying code in $G$ is a subset $C\subset V(G)$, with the property that $B_r(u)\cap C\neq B_r(v)\cap C$, for all distinct $u,v\in V(G)$ and $B_r(v)\cap C\neq\emptyset$, for all $v\in V(G)$. We study graphs with large symmetric differences and $(p,\beta)$-jumbled graphs and estimate the minimum size of a vertex-identifying code in each graph

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