Union-closed Sets Conjecture Research Articles

Improved Lower Bound for Frankl’s Union-Closed Sets Conjecture

Improved Lower Bound for Frankl’s Union-Closed Sets Conjecture

On the union-closed sets conjecture

We give a new proof of a theorem of Lo Faro concerning minimal counterexamples to the union-closed conjecture, and we prove some new results concerning the “ε-version” of the union-closed conjecture, and concerning the minimal maximal frequency in union-closed families.

Well-Graded Families and the Union-Closed Sets Conjecture

The union-closed sets conjecture states that if a finite family of sets $\mathcal{F}$ is union-closed, then there must be some element contained in at least half of the sets of $\mathcal{F}$. In this work we study the relationship between the union-closed sets conjecture and union-closed families that have the property of being well-graded. In doing so, we show how the density and other properties are affected by the extra structure contained in well-graded families, and we also give several conditions under which well-graded families satisfy the union-closed sets conjecture.

Extremal Union-Closed Set Families

A family of finite sets is called union-closed if it contains the union of any two sets in it. The Union-Closed Sets Conjecture of Frankl from 1979 states that each union-closed family contains an element that belongs to at least half of the members of the family. In this paper, we study structural properties of union-closed families. It is known that under the inclusion relation, every union-closed family forms a lattice. We call two union-closed families isomorphic if their corresponding lattices are isomorphic. Let $${{\mathcal {F}}}$$ be a union-closed family and $$\bigcup _{F\in {\mathcal {F}}} F$$ be the universe of $${\mathcal {F}}$$. Among all union-closed families isomorphic to $${{\mathcal {F}}}$$, we develop an algorithm to find one with a maximum universe, and an algorithm to find one with a minimum universe. We also study properties of these two extremal union-closed families in connection with the Union-Closed Set Conjecture. More specifically, a lower bound of sizes of sets of a minimum counterexample to the dual version of the Union-Closed Sets Conjecture is obtained.

Note on the Union-Closed Sets Conjecture

The union-closed sets conjecture states that if a finite family of sets $\mathcal{A} \neq \{\varnothing\}$ 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 set size of a union-closed family, $\mathcal{A}$, is at least $\frac{1}{2} \log_2 |\mathcal{A}|$. In order to do so, he showed that all union-closed families satisfy a particular condition, which in turn implies the preceding bound. Here, answering a question raised in the context of T. Gowers' polymath project on the union-closed sets conjecture, we show that Reimer's condition alone is not enough to imply that there is an element in at least half the sets.

A proof of Frankl’s union-closed sets conjecture for dismantlable lattices

In this paper, we prove Frankl’s Union-Closed Sets Conjecture for the class of dismantlable lattices, a more general class than the class of planar lattices. As a consequence of this result, we also prove that an upper semimodular lattice with breadth at most two satisfies the Conjecture.

New Conjectures for Union-Closed Families

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 prove that $2a$ is an upper bound to the number of sets in a union-closed family on a ground set of $n$ elements where each element is in at most $a$ sets for all $a,n\in \mathbb{N}^+$. Similarly, one could prove that the minimum number of sets containing the most frequent element in a (non-empty) union-closed family with $m$ sets and $n$ elements is at least $\frac{m}{2}$ for any $m,n\in \mathbb{N}^+$. Formulating these problems as integer programs, we observe that the optimal values we computed do not vary with $n$. We formalize these observations as conjectures, and show that they are not equivalent to the Frankl conjecture while still having wide-reaching implications if proven true. Finally, we prove special cases of the new conjectures and discuss possible approaches to solve them completely.

The Union-Closed Sets Conjecture for Small Families

We prove that the Union-Closed Sets Conjecture is true for separating union-closed families $$\mathcal {A}$$A with $$|\mathcal {A}| \le 2\left( m+\frac{m}{\log _2(m)-\log _2\log _2(m)}\right) $$|A|≤2m+mlog2(m)-log2log2(m) where m denotes the order of the universe $$\bigcup _{A\in \mathcal {A}} A$$źAźAA.

The Journey of the Union-Closed Sets Conjecture

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

The graph formulation of the union-closed sets conjecture

The union-closed sets conjecture asserts that in a finite non-trivial union-closed family of sets there has to be an element that belongs to at least half the sets. We show that this is equivalent to the conjecture that in a finite non-trivial graph there are two adjacent vertices each belonging to at most half of the maximal stable sets. In this graph formulation other special cases become natural. The conjecture is trivially true for non-bipartite graphs and we show that it holds also for the classes of chordal bipartite graphs, subcubic bipartite graphs, bipartite series-parallel graphs and bipartitioned circular interval graphs. We derive that the union-closed sets conjecture holds for all union-closed families being the union-closure of sets of size at most three.

A Computational Study for the Graph-theoretic Version of the Union-closed Sets Conjecture

ABSTRACT An induced subgraph S of a graph G is called a derived subgraph of G if S contains no isolated vertices. An edge e of G is said to be residual if e voccurs in more than half of the derived subgraphs of G . We prove some theorems which calculate the number of derived subgraphs for some special graphs. We also present a new algorithm SDSA that calculates the number of derived subgraphs for a given graph G and determines the residual and non-residual edges. Finally, we introduce a computational study which supports our results. Keywords 2 Union closed sets conjecture, induced graphs, derived subgraphs. 1. INTRODUCTION Example A union-closed family of sets A dis a finite collection of sets not all empty such that the union of any two members of A is also a member of A . The following Conjecture is due to Peter Frankl [1, 2, 3]. Conjecture 1 . Let A = { A 1 , A 2 , . . . , A n } be a union-closed family of n distinct sets. Then there exists an element which belongs to at least

On the Scope of Averaging for Frankl’s Conjecture

Let $\mathcal F$ be a union-closed family of subsets of an m-element set A. Let $n=|{\mathcal F}|\ge 2$ . For b ∈ A let w(b) denote the number of sets in $\mathcal F$ containing b minus the number of sets in $\mathcal F$ not containing b. Frankl’s conjecture from 1979, also known as the union-closed sets conjecture, states that there exists an element b ∈ A with w(b) ≥ 0. The present paper deals with the average of the w(b), computed over all b ∈ A. $\mathcal F$ is said to satisfy the averaged Frankl’s property if this average is non-negative. Although this much stronger property does not hold for all union-closed families, the first author (Czedli, J Comb Theory, Ser A, 2008) verified the averaged Frankl’s property whenever n ≥ 2 m − 2m/2 and m ≥ 3. The main result of this paper shows that (1) we cannot replace 2m/2 with the upper integer part of 2 m /3, and (2) if Frankl’s conjecture is true (at least for m-element base sets) and $n\ge 2^m-\lfloor 2^m/3\rfloor$ then the averaged Frankl’s property holds (i.e., 2m/2 can be replaced with the lower integer part of 2 m /3). The proof combines elementary facts from combinatorics and lattice theory. The paper is self-contained, and the reader is assumed to be familiar neither with lattices nor with combinatorics.

On averaging Frankl's conjecture for large union-closed-sets

Let F be a union-closed family of subsets of an m-element set A. Let n = | F | ⩾ 2 and for a ∈ A let s ( a ) denote the number of sets in F that contain a. Frankl's conjecture from 1979, also known as the union-closed sets conjecture, states that there exists an element a ∈ A with n − 2 s ( a ) ⩽ 0 . Strengthening a result of Gao and Yu [W. Gao, H. Yu, Note on the union-closed sets conjecture, Ars Combin. 49 (1998) 280–288] we verify the conjecture for the particular case when m ⩾ 3 and n ⩾ 2 m − 2 m / 2 . Moreover, for these “large” families F we prove an even stronger version via averaging. Namely, the sum of the n − 2 s ( a ) , for all a ∈ A , is shown to be non-positive. Notice that this stronger version does not hold for all union-closed families; however we conjecture that it holds for a much wider class of families than considered here. Although the proof of the result is based on elementary lattice theory, the paper is self-contained and the reader is not assumed to be familiar with lattices.

A graph-theoretic version of the union-closed sets conjecture

An induced subgraph S of a graph G is called a derived subgraph of G if S contains no isolated vertices. An edge e of G is said to be residual if e occurs in more than half of the derived subgraphs of G. We introduce the conjecture: Every non-empty graph contains a non-residual edge. This conjecture is implied by, but weaker than, the union-closed sets conjecture. We prove that a graph G of order n satisfies this conjecture whenever G satisfies any one of the conditions: δ(G) ≤ 2, log2n ≤ δ(G), n ≤ 10, or the girth of G is at least 6. Finally, we show that the union-closed sets conjecture, in its full generality, is equivalent to a similar conjecture about hypergraphs. © 1997 John Wiley & Sons, Inc. J Graph Theory 26: 155–163, 1997

A graph-theoretic version of the union-closed sets conjecture

A graph-theoretic version of the union-closed sets conjecture

A note on the union-closed sets conjecture

AbstractIt has been conjectured that for any union-closed set there exists some element which is contained in at least half the sets in . It is shown that this conjecture is true if the number of sets in is less than 25. Several conditions on a counterexample are also obtained.

A note of the union-closed sets conjecture

AbstractLet = {A1, …, An} be a union-closed set. This note establishes a property which must be possessed by any smallest counterexample to the Union-Closed Sets Conjecture. Specifically, a counterexample to the conjecture with minimal n has at least three distinct elements, each of which appears in exactly (n − 1)/2 of the .

On union-closed sets and Conway's sequence

In 1991 Renaud defined a boundary function φ(n) for union-closed sets, and evaluated it to n = 17. Also in 1991, Mallows examined a sequence a(n) defined recursively by Conway in 1988.Investigation of some properties of strictly reduced ordered power sets, a class of union-closed sets, leads to the conclusion that a(n + 1) is an upper bound for φ(n), and the union-closed sets conjecture holds if the conjecture φ(n) = a(n + 1) is valid.

Is the Union-closed sets conjecture the best possible?

AbstractA slightly strengthened version of the union-closed sets conjecture is proposed. It is shown that this version holds for a minimum set size of one or two and an examination of a boundary function shows that it holds for collections containing up to 19 sets. Some related conjectures are outlined.