Related papers: Generalized forbidden subposet problems

Given a symmetric monoidal category $C$ with product $\sqcup$, where the neutral element for the product is an initial object, we consider the poset of $\sqcup$-complemented subobjects of a given object $X$. When this poset has finite…

A quotient of a poset $P$ is a partial order obtained on the equivalence classes of an equivalence relation $\theta$ on $P$; $\theta$ is then called a congruence if it satisfies certain conditions, which vary according to different…

Let $\mathrm{G}$ be a subgroup of the symmetric group $\mathfrak S(U)$ of all permutations of a countable set $U$. Let $\overline{\mathrm{G}}$ be the topological closure of $\mathrm{G}$ in the function topology on $U^U$. We initiate the…

Let S be a basic closed semi-algebraic set in R^n and P the corresponding preordering in R[X_1,...,X_n]. We examine for which polynomials f there exist identities f+\ep q \in P for all \ep>0. These are precisely the elements of the…

We consider 'supersaturation' problems in partially ordered sets (posets) of the following form. Given a finite poset $P$ and an integer $m$ greater than the cardinality of the largest antichain in $P$, what is the minimum number of…

We say a finite poset $P$ is a tree poset if its Hasse diagram is a tree. Let $k$ be the length of the largest chain contained in $P$. We show that when $P$ is a fixed tree poset, the number of $P$-free set systems in $2^{[n]}$ is…

We show that a subset of $\mathbb{F}_{p}^{n}$ of $\mathrm{VC_{2}}$-dimension at most $k$ is well approximated by a union of atoms of a quadratic factor of complexity $(\ell,q)$ (denoting the complexities of the linear and quadratic part,…

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…

Let $\mathcal{F}_1$ and $\mathcal{F}_2$ be two families of subsets of an $n$-element set. We say that $\mathcal{F}_1$ and $\mathcal{F}_2$ are multiset-union-free if for any $A,B\in \mathcal{F}_1$ and $C,D\in \mathcal{F}_2$ the multisets…

The dimension of a poset $P$, denoted $\dim(P)$, is the least positive integer $d$ for which $P$ is the intersection of $d$ linear extensions of $P$. The maximum dimension of a poset $P$ with $|P|\le 2n+1$ is $n$, provided $n\ge2$, and this…

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…

We introduce $\delta$-cliffs, a generalization of permutations and increasing trees depending on a range map $\delta$. We define a first lattice structure on these objects and we establish general results about its subposets. Among them, we…

Given a finite poset $\mathcal P$, how small can a family $\mathcal F$ of subsets of $[n]$ be such that $\mathcal F$ does not contain an induced copy of $\mathcal P$, but $\mathcal F\cup\{X\}$ contains such a copy for all $X\in\mathcal…

For a family $\mathcal{F}$ of sets and a disjoint pair $A,B$ we let $\mathcal{F}(A,\overline{B})=\{F\in \mathcal{F}: A\subseteq F, ~B\cap F=\emptyset\}$. The \textbf{$(p,q)$-d\"omd\"od\"om} of a family $\mathcal{F}\subseteq 2^{[n]}$ is…

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…

A family $f_t(z)$ of polynomials over a number field $K$ will be called \emph{weighted homogeneous} if and only if $f_t(z)=F(z^e, t)$ for some binary homogeneous form $F(X, Y)$ and some integer $e\geq 2$. For example, the family $z^d+t$ is…

A family $\mathcal{F}$ of subsets of $\{1,2,\ldots,n\}$ is called a $t$-intersecting family if $|F\cap G| \geq t$ for any two members $F, G \in \mathcal{F}$ and for some positive integer $t$. If $t=1$, then we call the family $\mathcal{F}$…

Let $f(n)$ be the largest integer such that every poset on $n$ elements has a $2$-dimensional subposet on $f(n)$ elements. What is the asymptotics of $f(n)$? It is easy to see that $f(n)\geqslant n^{1/2}$. We improve the best known upper…

A collection of linear orders on $X$, say $\mathcal{L}$, is said to \emph{realize} a partially ordered set (or poset) $\mathcal{P} = (X, \preceq)$ if, for any two distinct $x,y \in X$, $x \preceq y$ if and only if $x \prec_L y$, $\forall L…

There is much recent interest in excluded subposets. Given a fixed poset $P$, how many subsets of $[n]$ can found without a copy of $P$ realized by the subset relation? The hardest and most intensely investigated problem of this kind is…