Related papers: The Union-Closed Sets Conjecture for Small Familie…
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 Union Closed Sets Conjecture states that in every finite, nontrivial set family closed under taking unions there is an element contained in at least half of all the sets of the family. We investigate two new directions with respect to…
The Frankl conjecture, also known as the union-closed sets conjecture, states that in any finite non-empty union-closed family, there exists an element in at least half of the sets. From an optimization point of view, one could instead…
The Union-Closed Sets Conjecture, also known as Frankl's conjecture, asks whether, for any union-closed set family $\mathcal{F}$ with $m$ sets, there is an element that lies in at least $\frac{1}{2}\cdot m$ sets in $\mathcal{F}$. In 2022,…
A family of sets is called union-closed if whenever $A$ and $B$ are sets of the family, so is $A\cup B$. The long-standing union-closed conjecture states that if a family of subsets of $[n]$ is union-closed, some element appears in at least…
Our aim in this note is to show that, for any $\epsilon>0$, there exists a union-closed family $\mathcal F$ with (unique) smallest set $S$ such that no element of $S$ belongs to more than a fraction $\epsilon$ of the sets in $\mathcal F$.…
A family of sets $\mathcal{A}$ is union-closed if it is finite and nonempty with member sets that are all finite and distinct (at least one of which is nonempty) and it satisfies the property $X, Y \in \mathcal{A} \implies X \cup Y \in…
The union-closed sets conjecture (Frankl's conjecture) says that for any finite union-closed family of finite sets, other than the family consisting only of the empty set, there exists an element that belongs to at least half of the sets in…
We prove Union-Closed sets conjecture.
The union-closed sets conjecture (Frankl's conjecture) says that for any finite union-closed family of finite sets, other than the family consisting only of the empty set, there exists an element that belongs to at least half of the sets in…
The Frankl or Union-Closed Sets conjecture states that for any finite union-closed family of sets $\mathcal{F}$ containing some nonempty set, there is some element $i$ in the ground set $U(\mathcal F) := \bigcup_{S \in \mathcal{F}} S$ of…
A family of sets is union-closed (UC) if the union of any two sets in the family is also in the family. Frankl's UC sets conjecture states that for any nonempty UC family $\mathcal{F} \subseteq 2^{[n]}$ such that $\mathcal{F} \neq…
We show that the biggest possible average set size in the complement $2^{\{1,2,\ldots, n\}} \setminus A$ of a union-closed family $A \subset 2^{\{1,2, \ldots, n\}}$ is $\tfrac{n+1}{2}$. With the same proof we get a sharp upper bound for the…
We provide a simple proof for the union-closed sets conjecture, a long-standing open problem in set theory with immediate applications to graph theory, number theory, and order-theory.
The Union Closed Sets Conjecture is one of the most renowned problems in combinatorics. Its appeal lies in the simplicity of its statement contrasted with the potential complexity of its resolution. The conjecture posits that, in any union…
Several results about the union-closed sets conjecture are presented.
We prove that the conjecture made by Peter Frankl in the late 1970s is true. In other words for every finite union-closed family which contains a non?empty set, there is an element that belongs to at least half of its m
A set system is called union closed if for any two sets in the set system their union is also in the set system. Gilmer recently proved that in any union closed set system some element belongs to at least a $0.01$ fraction of sets, and…
Let $\mathcal{A}$ be a union-closed family of sets with base set $b(\mathcal{A})=\bigcup_{A \in \mathcal{A}}A$ denoted by $[n]=\{1, \cdots, n\}$, and for any real $x>0$, let $\mathcal{A}_{<x} = \{A \in \mathcal{A} \ | \ |A| < x\}$. Also,…
For each given union-closed family F of n elements and m sets, we discuss the union-closed sets conjecture from height number of the UC family, which is a natural parameter from lattice theory. In this paper, we call it height number of…