Let F be a finite nonempty family of finite nonempty sets. We prove the following: (i) F satisfies the condition of the title if and only if for every pair of distinct subfamilies {A_1,...,A_r}, {B_1,...,B_s} of F, the union of the A_i is different from the union of the B_i. (ii) If F satisfies the condition of the title, then the number of subsets of the union of the members of F containing at least one set of F is odd. We give two applications of these results, one to number theory and one to commutative algebra.
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 $\mathcal{A}$.
In this very short paper, we point out that the average overlap density of a union-closed family $\mathcal{F}$ of subsets of $\{1,2,\ldots,n\}$ may be as small as $Θ((\log \log |\mathcal{F}|)/(\log |\mathcal{F}|))$, for infinitely many positive integers $n$.
Our aim in this note is to show that, for any $ε>0$, 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 $ε$ of the sets in $\mathcal F$. More precisely, we give an example of a union-closed family with smallest set of size $k$ such that no element of this set belongs to more than a fraction $(1+o(1))\frac{\log_2 k}{2k}$ of the sets in $\mathcal F$. We also give explicit examples of union-closed families containing `small' sets for which we have been unable to verify the Union-Closed Conjecture.
A finite family $\mathrsfs{F}$ of subsets of a finite set $X$ is union-closed whenever $f,g\in\mathrsfs{F}$ implies $f\cup g\in\mathrsfs{F}$. These families are well known because of Frankl's conjecture. In this paper we developed further the connection between union-closed families and upward-closed families started in Reimer (2003) using rising operators. With these techniques we are able to obtain tight lower bounds to the average of the length of the elements of $\mathrsfs{F}$ and to prove that the number of joint-irreducible elements of $\mathrsfs{F}$ can not exceed $2{n\choose \lfloor n/…
Given a union-closed family $\mathcal{F}$ of subsets of the universe $[n]$, with $\mathcal{F}$ not equal to the power set of $[n]$, a new subset $A$ can be added to it such that the resulting family remains union-closed. We construct a new family $\overline{\mathcal{F}}$ by adding to $\mathcal{F}$ all such $A$'s, and call this the closure of $\mathcal F$. This paper is dedicated to the study of various properties of such closures, including characterizing families whose closures equal the power set of $[n]$, providing a criterion for the existence of closure roots of such families etc.
In this article we will describe an algorithm to constructively enumerate non-isomorphic Union closed Sets and Moore sets. We confirm the number of isomorphism classes of Union closed Sets and Moore sets on n<=6 elements presented by other authors and give the number of isomorphism classes of Union closed Sets and Moore sets on 7 elements. Due to the enormous growth of the number of isomorphism classes it seems unlikely that constructive enumeration for 8 or more elements will be possible in the foreseeable future.
This paper discusses the question of how many non-empty subsets of the set $[n] = \{ 1, 2, ..., n\}$ we can choose so that no chosen subset is the union of some other chosen subsets. Let $M(n)$ be the maximum number of subsets we can choose. We construct a series of such families, which leads to lower bounds on $M(n)$. We also give upper bounds on $M(n)$. Finally, we propose several conjectures on the tightness of our lower bound for $M(n)$.
A family of sets is union-free if there are no three distinct sets in the family such that the union of two of the sets is equal to the third set. Kleitman proved that every union-free family has size at most $(1+o(1))\binom{n}{n/2}$. Later, Burosch--Demetrovics-Katona-Kleitman-Sapozhenko asked for the number $α(n)$ of such families, and they proved that $2^{\binom{n}{n/2}}\leq α(n) \leq 2^{2\sqrt{2}\binom{n}{n/2}(1+o(1))}$. They conjectured that the constant $2\sqrt{2}$ can be removed in the exponent of the right hand side. We prove their conjecture by formulating a new container-type theorem…
We show that there is some absolute constant $c>0$, such that for any union-closed family $\mathcal{F} \subseteq 2^{[n]}$, if \mbox{$|\mathcal{F}| \geq (\frac{1}{2}-c)2^n$}, then there is some element $i \in [n]$ that appears in at least half of the sets of $\mathcal{F}$. We also show that for any union-closed family $\mathcal{F} \subseteq 2^{[n]}$, the number of sets which are not in $\mathcal{F}$ that cover a set in $\mathcal{F}$ is at most $2^{n-1}$, and provide examples where the inequality is tight.
Let F be a finite union-closed family of sets whose largest set contains n elements. In \cite{Wojcik92}, Wojcik defined the density of F to be the ratio of the average set size of F to n and conjectured that the minimum density over all union-closed families whose largest set contains n elements is (1 + o(1))\log_2(n)/(2n) as n approaches infinity. We use a result of Reimer \cite{Reimer03} to show that the density of F is always at least log_2(n)/(2n), verifying Wojcik's conjecture. As a corollary we show that for n \geq 16, some element must appear in at least \sqrt{(\log_2(n))/n}(|F|/2) sets…
We present equivalent formulations for concepts related to set families for which every subfamily with empty intersection has a bounded sub-collection with empty intersection. Hereby, we summarize the progress on the related questions about the maximum size of such families. In this work we solve a boundary case of a problem of Tuza for non-trivial $q$-Helly families, by applying Karamata's inequality and determining the minimum size of a $2$-self-centered graph for which the common neighborhood of every pair of vertices contains a clique of size $q-2$.
Let G be a graph with vertices V and edges E. Let F be the union-closed family of sets generated by E. Then F is the family of subsets of V without isolated points. Theorem: There is an edge e belongs to E such that |{U belongs to F | e belongs to U}| =< 1/2|F|. This is equivalent to the following assertion: If H is a union-closed family generated by a family of sets of maximum degree two, then there is an $x$ such that |{U belongs to H | x belongs to U}| > 1/2|H|. This is a special case of the union-closed sets conjecture. To put this result in perspective, a brief overview of research on the…
Let $A\subset \mathbb{N}^{n}$ be an $r$-wise $s$-union family, that is, a family of sequences with $n$ components of non-negative integers such that for any $r$ sequences in $A$ the total sum of the maximum of each component in those sequences is at most $s$. We determine the maximum size of $A$ and its unique extremal configuration provided (i) $n$ is sufficiently large for fixed $r$ and $s$, or (ii) $n=r+1$.
We give a simple argument showing that the number of edges in the intersection graph $G$ of a family of $n$ sets in the plane with a linear union-complexity is $O(ω(G)n)$. In particular, we prove $χ(G)\leq \text{col}(G)< 19ω(G)$ for intersection graph $G$ of a family of pseudo-discs, which improves a previous bound.
Let R be a family of n axis-parallel rectangles with packing number p-1, meaning that among any p of the rectangles, there are two with a non-empty intersection. We show that the union complexity of R is at most O(n+p^2), and that the (<=k)-level complexity of R is at most O(kn+k^2p^2). Both upper bounds are tight.
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 average frequency in complements of union-closed families. This implies an asymptotic version of the union-closed sets conjecture, formulated in terms of complements of union-closed families.
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 half the sets of the family. A natural weakening is that the union-closed conjecture holds for large families; that is, families consisting of at least $p_02^n$ sets for some constant $p_0$. The first result in this direction appears in a recent paper of Balla, Bollobás and Eccles \cite{BaBoEc}, who showed that union-closed families of at least $\frac{2}{3}…
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 $A\uplus C$ and $B\uplus D$ are different, unless both $A = B$ and $C= D$. We derive a new upper bound on the maximal sizes of multiset-union-free pairs, improving a result of Urbanke and Li.
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 $\mathcal{F} \neq \left\{\emptyset\right\}$, there exists an element $i \in [n]$ that is contained in at least half the sets of $\mathcal{F}$. The 3-sets conjecture of Morris states that the smallest number of distinct 3-sets (whose union is an $n$-set) that ensure Frankl's conjecture is satisfied for any UC family that contains them is $ \lfloor{n/2\rfloor} + 1$ for all $n \geq 4$. F…
We prove lower bounds for the fraction of edges of an $r$-graph which can be covered by the union of $k$ 1-factors. The special case $r=3$ yields some known results for cubic graphs. Furthermore, we introduce the concept of $k$-overfull-free $r$-graphs and achieve better bounds for these graphs.
Let F be a family of subsets of an n-element set not containing four distinct members such that A union B is contained in C intersect D. It is proved that the maximum size of F under this condition is equal to the sum of the two largest binomial coefficients of order n. The maximum families are also characterized. A LYM-type inequality for such families is given, too.
A family $\F$ of sets is said to be intersecting if any two sets in $\F$ have nonempty intersection. The celebrated Erd{\H o}s-Ko-Rado theorem determines the size and structure of the largest intersecting family of $k$-sets on an $n$-set $X$. An $(s,t)$-union intersecting family is a family of $k$-sets on an $n$-set $X$ such that for any $A_1,\ldots,A_{s+t}$ in this family, $\left(\cup_{i=1}^sA_i\right)\cap\left(\cup_{i=1}^t A_{i+s}\right)\neq \varnothing.$ Let $\ell(\F)$ be the minimum number of sets in $\F$ such that by removing them the resulting subfamily is intersecting. In this paper, fo…
We focus our attention on well-covered graphs that are vertex decomposable. We show that for many known families of these vertex decomposable graphs, the set of shedding vertices forms a dominating set. We then construct three new infinite families of well-covered graphs, none of which have this property. We use these results to provide a minimal counterexample to a conjecture of Villarreal regarding Cohen-Macaulay graphs.
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 $\…
Let Omega be a finite set and let S be a set system on Omega. For x in Omega, we denote by d_{S}(x) the number of members of S containing x. A long-standing conjecture of Frankl states that if S is union-closed then d(x) \geq |S|/2 for some x in Omega. We consider a related question. Define the weight of S to be w(S)= \sum_{A in S} |A|. Suppose S is union-closed. How small can w(S) be? Reimer showed that w(S) \geq |S| \log_{2} |S| /2, and that this inequality is sharp. In this paper we show how his bound may be improved if we have some additional information about the domain Omega of S: if S s…
We give two constructions of strongly regular Cayley graphs on finite fields $\F_q$ by using union of cyclotomic classes and index 2 Gauss sums. In particular, we obtain twelve infinite families of strongly regular graphs with new parameters.
A family of sets is said to be \emph{intersecting} if any two sets in the family have nonempty intersection. In 1973, Erdős raised the problem of determining the maximum possible size of a union of $r$ different intersecting families of $k$-element subsets of an $n$-element set, for each triple of integers $(n,k,r)$. We make progress on this problem, proving that for any fixed integer $r \geq 2$ and for any $k \leq (\tfrac{1}{2}-o(1))n$, if $X$ is an $n$-element set, and $\mathcal{F} = \mathcal{F}_1 \cup \mathcal{F}_2 \cup \ldots \cup \mathcal{F}_r$, where each $\mathcal{F}_i$ is an intersecti…
A graph G is well-covered if all its maximal independent sets are of the same cardinality. Assume that a weight function w is defined on its vertices. Then G is w-well-covered if all maximal independent sets are of the same weight. For every graph G, the set of weight functions w such that G is w-well-covered is a vector space. Given an input graph G without cycles of length 4, 5, and 6, we characterize polynomially the vector space of weight functions w for which G is w-well-covered. Let B be an induced complete bipartite subgraph of G on vertex sets of bipartition B_{X} and B_{Y}. Assume tha…
Union-free families of subsets of $[n]=\{1,... n\}$ have been studied in \cite{FF}. In this paper, we provide a complete characterization of maximal {\it symmetric difference}-free families of subsets of $[n]$.
Recently, Earl, Vander Meulen, and Van Tuyl characterized some families of Cohen-Macaulay or Buchsbaum circulant graphs discovered by Boros-Gurvich-Milani$\check{\text{c}}$, Brown-Hoshino, and Moussi. In this paper, we will characterize those families of circulant graphs which satisfy Serre's condition $S_2$. More precisely, we show that for some families of circulant graphs, $S_2$ property is equivalent to well-coveredness or Buchsbaumness, and for some other families it is equivalent to Cohen-Macaulayness. We also give examples of infinite families of circulant graphs which are Buchsbaum but…
Two perfect matchings $P$ and $Q$ of the complete graph on $2k$ vertices are said to be set-wise $t$-intersecting if there exist edges $P_{1}, \cdots, P_{t}$ in $P$ and $Q_{1}, \cdots, Q_{t}$ in $Q$ such that the union of edges $P_{1}, \cdots, P_{t}$ has the same set of vertices as the union of $Q_{1}, \cdots, Q_{t}$ has. In this paper we prove an extension of the famous Erdős-Ko-Rado (EKR) theorem to set-wise $2$-intersecting families of perfect matching on all values of $k$, and we conjecture similar statement for all $t\geq 2$.
A family of sets F is said to be union-closed if A \cup B is in F for every A and B in F. Frankl's conjecture states that given any finite union-closed family of sets, not all empty, there exists an element contained in at least half of the sets. Here we prove that the conjecture holds for families containing three 3-subsets of a 5-set, four 3-subsets of a 6-set, or eight 4-subsets of a 6-set, extending work of Poonen and Vaughan. As an application we prove the conjecture in the case that the largest set has at most nine elements, extending a result of Gao and Yu. We also pose several open que…
We prove that there exist bipartite Ramanujan graphs of every degree and every number of vertices. The proof is based on analyzing the expected characteristic polynomial of a union of random perfect matchings, and involves three ingredients: (1) a formula for the expected characteristic polynomial of the sum of a regular graph with a random permutation of another regular graph, (2) a proof that this expected polynomial is real rooted and that the family of polynomials considered in this sum is an interlacing family, and (3) strong bounds on the roots of the expected characteristic polynomial o…
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 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 t…
Motivated by the concept of well-covered graphs, we define a graph to be well-bicovered if every vertex-maximal bipartite subgraph has the same order (which we call the bipartite number). We first give examples of them, compare them with well-covered graphs, and characterize those with small or large bipartite number. We then consider graph operations including the union, join, and lexicographic and cartesian products. Thereafter we consider simplicial vertices and 3-colored graphs where every vertex is in triangle, and conclude by characterizing the maximal outerplanar graphs that are well-bi…
Let G be a family of n-vertex graphs of uniform degree 2 with the property that the union of any two member graphs has degree four. We determine the leading term in the asymptotics of the largest cardinality of such a family. Several analogous problems are discussed.
Resolving Erdős-Ulam Monochromatic Union-Closed Family Conjectures
Electro-Gravitational Space Propulsion Laboratory (EGSPL)（电引力空间推进实验室）；Maulana Azad National Institute of Technology Bhopal（莫拉纳·阿扎德国家技术学院博帕尔分校）；National Taiwan University（国立台湾大学）
Motivated by a question on the maximal number of vertex disjoint Schrijver graphs in the Kneser graph, we investigate the following function, denoted by $f(n,k)$: the maximal number of Hamiltonian cycles on an $n$ element set, such that no two cycles share a common independent set of size more than $k$. We shall mainly be interested in the behavior of $f(n,k)$ when $k$ is a linear function of $n$, namely $k=cn$. We show a threshold phenomenon: there exists a constant $c_t$ such that for $c<c_t$, $f(n,cn)$ is bounded by a constant depending only on $c$ and not on $n$, and for $c_t <c$, $f(n,cn)…
