Related papers: Standard Examples as Subposets of Posets

We prove that every planar poset $P$ of height $h$ has dimension at most $192h + 96$. This improves on previous exponential bounds and is best possible up to a constant factor. We complement this result with a construction of planar posets…

For k>=3 let A \subset [1,N] be a set not containing a solution to a_1 x_1+...+a_k x_k=a_1 x_{k+1}+...+a_k x_{2k} in distinct integers. We prove that there is an epsilon>0 depending on the coefficients of the equation such that every such A…

Given $s,t\in\mathbb{N}$, a complete bipartite poset $\mathcal{K}_{s,t}$ is a poset whose Hasse diagram consists of $s$ pairwise incomparable vertices in the upper layer and $t$ pairwise incomparable vertices in the lower layer, such that…

Fix a density d in (0,1], and let F_p^n be a finite field, where we think of p fixed and n tending to infinity. Let S be any subset of F_p^n having the minimal number of three-term progressions, subject to the constraint |S| is at least…

Let $P(n)$ be the set of all posets with $n$ elements. Let $P^{(j)}(n)$, $1\leq j\leq 2^n,$ be the number of all posets with $n$ elements possessing exactly $j$ antichains. We have determined the numbers $P^{(j)}(7),$ $1\leq j\leq 128$, and…

Let $k \geq 1$ be an integer. A set $A \subset \mathbb{Z}$ is a $k$-fold Sidon set if $A$ has only trivial solutions to each equation of the form $c_1 x_1 + c_2 x_2 + c_3 x_3 + c_4 x_4 = 0$ where $0 \leq |c_i | \leq k$, and $c_1 + c_2 + c_3…

The aim of this paper is to prove that if a planar set $A$ has a difference set $\Delta(A)$ satisfying $\Delta(A)\subset \Z^++s$ for suitable $s$ than $A$ has at most 3 elements. This result is motivated by the conjecture that the disk has…

The number of embeddings of a partially ordered set $S$ in a partially ordered set $T$ is the number of subposets of $T$ isomorphic to $S$. If both, $S$ and $T$, have only one unique maximal element, we define good embeddings as those in…

We show that the separative quotient of the poset (P(L),\subset) of isomorphic suborders of a countable scattered linear order L is \sigma-closed and atomless. So, under the CH, all these posets are forcing-equivalent (to P(\omega)/Fin).

We introduce so-called consistent posets which are bounded posets with an antitone involution ' where the lower cones of x,x' and of y,y' coincide provided x,y are different form 0,1 and, moreover, if x,y are different form 0 then their…

We give a construction of an infinite set of points $A$ in $\mathbb{R}^2$ such that any subset $P\subseteq A$ has a constant density subset $P'$ with no three points collinear and yet $A$ cannot be separated into finitely many subsets such…

Order dimension theory measures the complexity of partially ordered sets by quantifying how far they are from being linearly ordered. In this paper we study classical bounding results for order dimension within the framework of reverse…

In this paper, we study the posets of classes of subgroups of finite group having same set of orders of elements. We show that this poset is a chain only in the case of p-groups and moreover, we characterize all finite groups for which this…

Experience shows that the poset of levels (or dimensions) of the topos of presheaves on some elegant Reedy categories may be equipped with a monotone increasing `successor' function which, as the case of simplicial sets shows, is different…

Let $(P,\leq)$ be a finite poset (partially ordered set), where $P$ has cardinality $n$. Consider linear extensions of $P$ as permutations $x_1x_2\cdots x_n$ in one-line notation. For distinct elements $x,y\in P$, we define…

For each poset $H$ whose Hasse diagram is a tree of height $k$, we show that the largest size of a family $\cF$ of subsets of $[n]=\{1,..., n\}$ not containing $H$ as an induced subposet is asymptotic to $(k-1){n\choose \fl{n/2}}$. This…

We show that height $h$ posets that have planar cover graphs have dimension $\mathcal{O}(h^6)$. Previously, the best upper bound was $2^{\mathcal{O}(h^3)}$. Planarity plays a key role in our arguments, since there are posets such that (1)…

If, for a subset S of Z^k, we compare the conditions of being parametrizable (a) by a single k-tuple of polynomials with integer coefficients, (b) by a single k-tuple of integer-valued polynomials and, (c) by finitely many k-tuples of…

We prove a simple formula for arbitrary cluster variables in the marked surfaces model. As part of the formula, we associate a labeled poset to each tagged arc, such that the associated $F$-polynomial is a weighted sum of order ideals. Each…

We show that any finite $S \subset \mathbb{R}^d$ in general position has arbitrarily large supersets $T \supseteq S$ in general position with the property that $T$ contains no empty convex polygon, or hole, with $C_d$ points, where $C_d$ is…