Related papers: On the $1/3-2/3$ Conjecture

In this paper, we prove the twin prime conjecture showing that \begin{align} \sum \limits_{\substack{p\leq x\\p,p+2\in \mathbb{P}}}1\geq (1+o(1))\frac{x}{2\mathcal{C}\log^2 x}\nonumber \end{align} where $\mathcal{C}:=\mathcal{C}(2)>0$ fixed…

This paper first gives a necessary and sufficient condition that a lattice $L$ can be represented as the collection of all up-sets of a poset. Applying the condition, it obtains a necessary and sufficient condition that a lattice can be…

The union-closed sets conjecture states that if a family of sets $\mathcal{A} \neq \{\emptyset\}$ is union-closed, then there is an element which belongs to at least half the sets in $\mathcal{A}$. In 2001, D. Reimer showed that the average…

The notion of (3+1)-avoidance has shown up in many places in enumerative combinatorics. The natural goal of enumeration of all (3+1)-avoiding posets remains open. In this paper, we enumerate graded (3+1)-avoiding posets for both reasonable…

We introduce partially ordered sets (posets) with an additional structure given by a collection of vector subspaces of an algebra $A$. We call them algebraically equipped posets. Some particular cases of these, are generalized equipped…

Talagrand conjectured that if a family of sets $\mathcal{F}$ over $X = \{ 1,2,\cdots, N \}$ is of large measure, then constant times of unions of sets in $\mathcal{F}$ will cover a large portion of the power set of $X$. This conjecture is a…

In the theory of partially-ordered sets, the two-dimensional Boolean lattice is known as the diamond. In this paper, we show that, if $\mathcal{F}$ is a family in the $n$-dimensional Boolean lattice that has no diamond as a subposet, then…

For a (finite) partially ordered set (poset) $P$, we call a dominating set $D$ in the comparability graph of $P$, an order-sensitive dominating set in $P$ if either $x\in D$ or else $a<x<b$ in $P$ for some $a,b\in D$ for every element $x$…

We address a special case of a conjecture of M. Talagrand relating two notions of "threshold" for an increasing family $\mathcal F$ of subsets of a finite set $V$. The full conjecture implies equivalence of the "Fractional…

Recently, Kitaev and Remmel posed a conjecture concerning the generating function for the number of unlabeled $(2+2)$-free posets with respect to number of elements and number of minimal elements. In this paper, we present a combinatorial…

Consider the recursive relation generating a new positive integer $n_{\ell +1}$ from the positive integer $n_{\ell }$ according to the following simple rules: if the integer $n_{\ell }$ is odd, $n_{\ell +1}=3n_{\ell }+1$; if the integer…

We study three different poset structures on the set of all compositions. In the first case, the covering relation consists of inserting a part of size one to the left or to the right, or increasing the size of some part by one. The…

We prove that if $\leq$ is an analytic partial order then either $\leq$ can be extended to a (boldface) $\Delta^1_2$ linear order similar to an antichain in $2^{<\omega_1}$ ordered lexicographically or a certain Borel partial order $\leq_0$…

A partially ordered pattern (abbreviated POP) is a partially ordered set (poset) that generalizes the notion of a pattern when we are not concerned with the relative order of some of its letters. The notion of partially ordered patterns…

Tukey order are used to compare the cofinal complexity of partially order sets (posets). We prove that there is a $2^\mathfrak{c}$-sized collection of sub-posets in $2^\omega$ which forms an antichain in the sense of Tukey ordering. Using…

Let n be an even positive integer and F be the field \GF(2). A word in F^n is called balanced if its Hamming weight is n/2. A subset C \subseteq F^n$ is called a balancing set if for every word y \in F^n there is a word x \in C such that y…

Given a finite poset $P$, we consider the largest size $\lanp$ of a family $\F$ of subsets of $[n]:=\{1,...,n\}$ that contains no subposet $P$. This continues the study of the asymptotic growth of $\lanp$; it has been conjectured that for…

Given a finite poset $\mathcal P$, we say that a family $\mathcal F$ of subsets of $[n]$ is $\mathcal P$-saturated if $\mathcal F$ does not contain an induced copy of $\mathcal P$, but adding any other set to $\mathcal F$ creates an induced…

In 1971 Trotter conjectured that every finite poset on at least $3$ points has a pair whose removal does not decrease the dimension by more than $1$. In 1992 Brightwell and Scheinerman introduced fractional dimension of posets, and they…

We show that the theories of partially ordered sets, lattices, semilattices, Boolean algebras, Heyting algebras with a further coarser partial order, or a linearization, or an auxiliary relation have the strong amalgamation property,…