Related papers: On the Union-Closed Sets Conjecture

We prove Union-Closed sets conjecture.

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.

We survey the state of the union-closed sets conjecture.

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…

We show that the Union-Closed Conjecture holds for the union-closed family generated by the cyclic translates of any fixed set.

The Frankl conjecture (called also union-closed sets conjecture) is one of the famous unsolved conjectures in combinatorics of finite sets. In this short note, we introduce and to some extent justify some variants of the Frankl 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…

We prove that the union-closed sets conjecture is true for separating union-closed families $\mathcal{A}$ with $|\mathcal{A}| \leq 2\left(m+\frac{m}{\log_2(m)-\log_2\log_2(m)}\right)$ where $m$ denotes the number of elements in…

In this very short note, we give a counterexample to a recent conjecture of Gilmer which would have implied the union-closed conjecture.

We provide a proof of the union-closed sets conjecture, by means of a suitable refinement of the breakthrough entropy-approach introduced by Gilmer. The novelty here is to consider a convex combination of $A$ and $A\cup B$, where $A,B$ are…

The union-closed sets conjecture, also known as Frankl's conjecture, is a well-studied problem with various formulations. In terms of lattices, the conjecture states that every finite lattice $L$ with more than one element contains a…

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…

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…

We give some results and conjectures about recurrence relations for certain sequences of binomial sums.

We survey most of the known results concerning the Eisenbud-Green-Harris Conjecture. Our presentation includes new proofs of several theorems, as well as a unified treatment of many results which are otherwise scattered in the literature.…

The union-closed sets conjecture, attributed to P\'eter Frankl from 1979, states that for any non-empty finite union-closed family of finite sets not consisting of only the empty set, there is an element that is in at least half of the sets…

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

In this note, we present a conjecture on intersections of set families, and a rephrasing of the conjecture in terms of principal downsets of Boolean lattices. The conjecture informally states that, whenever we can express the measure of a…

We prove the validity of the strong version of the union of uniform closed balls conjecture, formulated in 2011 as [4, Conjecture 2.5], in the plane.

The union-closed sets conjecture (sometimes referred to as Frankl's conjecture) states that every finite, nontrivial union-closed family of sets has an element that is in at least half of its members. Although the conjecture is known to be…