Union-closed families with small average overlap densities

Generation of Union Closed Sets and Moore families

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.

Union-Free Families of Subsets

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 $M(n)$.

On the number of union-free families

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…

Two Results on Union-Closed Families

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.

Minimum density of union-closed families

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…

Set systems without a simplex, Helly hypergraphs and union-efficient families

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 $q-2$.

链条件与一般并集闭家族中的最优元素

Chain Conditions and Optimal Elements in Generalized Union-Closed Families of Sets

本文通过引入链条件和最优元素，证明在特定条件下并集闭猜想对有限和无限家族均成立，并给出了非并集闭的家族示例。

Graph generated union-closed families of sets

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…

对称并闭族

Symmetric Union Closed Families

本文证明具有特定对称性的图及其相关图类（如圆柱网格图和环面网格图）满足并闭猜想，并利用对称性论证简化了循环平移生成并闭族的已知证明，扩展了满足该猜想的并闭族范围。

Multiply union families in $\mathbb{N}^n$

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) $n=r+1$.

并闭族的熵界与全局耦合

Entropy bounds and global couplings for union-closed families

通过计算机辅助证明，结合独立与条件独立采样及全局耦合，证明了有限并闭族中某元素出现频率至少为0.38288525，并给出依赖族大小的额外频率界。

An asymptotic version of the union-closed sets conjecture

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 stability result for the union-closed size problem

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}…

并集闭族中的局部构型

Local Configurations in Union-Closed Families

本文研究了并集闭集猜想，求得了最小整数$FC(k,n)$的新值与界限，回答了Vaughan等人的问题，并通过推广Poonen定理引入局部构型判据，为许多先前未知的集族类别证明了该猜想。

并闭集族中的频繁元素

Frequent elements in union-closed set families

本文结合 Gilmer 的熵方法与 Knill 的组合论证，证明了 Nagel 提出的关于并闭集族中第 k 个最常见元素出现频率的推广猜想，并刻画了取等集族及无穷极限下的渐近下界。

非均匀$t$-相交族的稳定性

Stabilities for non-uniform $t$-intersecting families

本文在极值组合学的集合相交问题领域，进一步拓展$s$-并族的极值结果，确定了既不包含于Katona族也不包含于Frankl族的$s$-并族的最大规模，并刻画了所有达到上界的极值族。

关于并集封闭集猜想的注记

Notes on the Union Closed Sets Conjecture

本文研究了并集封闭集猜想的两个新方向：探讨集族中所有元素的频率以提出推广问题，并研究等价结构以得到猜想的弱化版本，同时提出了新的开放问题。

Largest family without A union B contained in C intersect D

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.

Size and structure of large $(s,t)$-union intersecting families

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…

Note on the number of edges in families with linear union-complexity

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.

On the union complexity of families of axis-parallel rectangles with a low packing number

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.

Frankl 关于交叉并族猜想的证明

A proof of Frankl's conjecture on cross-union families

本文研究交叉并子集族的平均规模上界，证明 Frankl 猜想的强形式：在相应条件下，算术平均的唯一极值结构是所有族均等同于 $[n-1]$ 的 $k$ 元子集族。

关于并闭集族第二频率的进一步分析

Further analysis on the second frequency of union-closed set families

本文针对 Nagel 关于并闭集族元素频率的强化猜想，利用线性规划进一步刻画 k=2 时集族不满足该猜想的情形。

Families of finite sets in which no set is covered by the union of the others

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.

相交并集族的卢贝尔界

The binomial norm of intersecting-union families

研究关于满足特定条件的族\(\mathcal{F}\subseteq 2^{[n]}\) 的卢贝尔界问题，核心方法是用\(p\)偏差测度框架取代循环置换论证并积分，主要贡献是证明弗兰克尔猜想并确定极值族，改进了圆法的估计。

Minimal weight in union-closed families

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…

On the union of intersecting families

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…

Characterizing 3-sets in Union-Closed Families

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…

On the sum of sizes of overlapping families

Let $\mathcal{A}_1,\ldots,\mathcal{A}_m$ be families of $k$-subsets of an $n$-set. Suppose that one cannot choose pairwise disjoint edges from $s+1$ distinct families. Subject to this condition we investigate the maximum of $|\mathcal{A}_1|+\ldots+|\mathcal{A}_m|$. Note that the subcase $m=s+1$, $\mathcal{A}_1=\ldots=\mathcal{A}_m$ is the Erdős Matching Conjecture, one of the most important open problems in extremal set theory. We provide some upper bounds, a general conjecture and its solution for the range $n\geq 4k^2s$.

并闭集合中的同构

Isomorphism in Union-Closed Sets

本文证明纯并闭族之间的同构可由超同构实现，从而说明并闭族可由其格结构在同构意义下唯一重构，为 Frankl 并闭集合猜想提供新视角。

关于并闭集族补集的立方视角

A Cubical Perspective on Complements of Union-Closed Families of Sets

本文从立方集视角研究简单根集族，为有限集子集族构造自然立方集，证明包含空集的简单根集族对应的立方集无环，并利用 Euler-Poincarè 公式导出满足该族的一个公式及初等证明。

Suboptimal $s$-union familes and $s$-union antichains for vector spaces

Let $V$ be an $n$-dimensional vector space over the finite field $\mathbb{F}_{q}$, and let $\mathcal{L}(V)=\bigcup_{0\leq k\leq n}\left[V\atop k\right]$ be the set of all subspaces of $V$. A family of subspaces $\mathcal{F}\subseteq \mathcal{L}(V)$ is $s$-union if dim$(F+F')\leq s$ holds for all $F$, $F'\in\mathcal{F}$. A family $\mathcal{F}\subseteq \mathcal{L}(V)$ is an antichain if $F\nleq F'$ holds for any two distinct $F, F'\in \mathcal{F}$. The optimal $s$-union families in $\mathcal{L}(V)$ have been determined by Frankl and Tokushige in $2013$. The upper bound of cardinalities of $s$-un…

Maximal Symmetric Difference-Free Families of Subsets of $[n]$

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 $[n]$.

集合的并闭族的一个平均结果

An averaging result for union-closed families of sets

本文研究了集合并闭族的平均大小问题，证明了在可分且高度为4、最小不可约子族大小不超过2的条件下，成员集合的平均大小至少为 $n/2$，并验证了该高度界限的最大性。

An Upper Bound on the Sizes of Multiset-Union-Free Families

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.

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 $\…

关于具有给定数量的 $k$-集合的并封闭族

On union-closed families with prescribed number of $k$-sets

本文研究并封闭族中大小至少为 n 的成员最小数量问题，证明 Leck--Roberts--Simpson 加权猜想在多个条带和条件下成立，并给出充分界。

FC-families, and improved bounds for Frankl's Conjecture

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…

Interlacing Families IV: Bipartite Ramanujan Graphs of All Sizes

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…

关于并闭集族的猜想

Conjectures on union-closed families of sets

本文研究了并闭集族猜想类 UC_x，证明了当 x 属于特定区间时该猜想成立，并探讨了 Frankl 猜想的两个加强形式。

Note on the union-closed sets conjecture

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…

关于在多个完全子图的并集中的 $r$-独立集的希尔顿-米勒定理

Hilton-Milner Theorem for the $r$-independent sets in a union of cliques

本文提出了一种针对多个完全子图并集中的 $r$-独立集的希尔顿-米勒定理，扩展了之前对深度二爪图的研究，验证了Holroyd--Talbot猜想。

Intersecting families of extended balls in the Hamming spaces

A family $\mathcal{F}$ of subsets of a set $X$ is $t$-intersecting if $\vert A_i \cap A_j \vert \geq t$ for every $A_i, \; A_j \in \mathcal{F}$. We study intersecting families in the Hamming geometry. Given $X=\mathbb{F}_q^3$ a vector space over the finite field $\mathbb{F}_q$, consider a family where each $A_i$ is an extended ball, that is, $A_i$ is the union of all balls centered in the scalar multiples of a vector. The geometric behavior of extended balls is discussed. As the main result, we investigate a ``large" arrangement of vectors whose extended balls are ``highly intersecting". Consi…

Cross-intersecting subfamilies of levels of hereditary families

A set $A$ $t$-intersects a set $B$ if $A$ and $B$ have at least $t$ common elements. Families $\mathcal{A}_1, \mathcal{A}_2, \dots, \mathcal{A}_k$ of sets are cross-$t$-intersecting if, for every $i$ and $j$ in $\{1, 2, \dots, k\}$ with $i \neq j$, each set in $\mathcal{A}_i$ $t$-intersects each set in $\mathcal{A}_j$. An active problem in extremal set theory is to determine, for a given finite family $\mathcal{F}$, the structure of $k$ cross-$t$-intersecting subfamilies whose sum or product of sizes is maximum. For a family $\mathcal{H}$, the $r$-th level $\mathcal{H}^{(r)}$ of $\mathcal{H}$ …

无小集合的并闭族中丰富元素的数量

The number of abundant elements in union-closed families without small sets

本文研究并闭族中丰富元素的数量，证明在最小集合大小 k 接近最大集合大小 n 时至少有 k 或 k-1 个丰富元素，并构造出恰好达到这些下界的反例，同时阐明 Cui-Hu 猜想与 Frankl、Poonen 猜想之间的关系。

关于加权立方体上并闭猜想的部分结果

Partial results for union-closed conjectures on the weighted cube

本文通过在布尔立方体上引入一般概率分布，提出了并闭猜想的推广形式，并给出了 Karpas 特例和 Knill 对数下界的广义结果。

解决 Erdős-Ulam 单色并封闭族猜想

Resolving Erdős-Ulam Monochromatic Union-Closed Family Conjectures

本文证明了 Erdős-Ulam 关于单色并封闭族的两个猜想：任何二染色中均存在大小超多项式增长的单色并封闭族，且某些染色无指数大小族，结果对任意颜色数成立并已在 Lean 中验证。

Structural results for conditionally intersecting families and some applications

Let $k\ge d\ge 3$ be fixed. Let $\mathcal{F}$ be a $k$-uniform family on $[n]$. Then $\mathcal{F}$ is $(d,s)$-conditionally intersecting if it does not contain $d$ sets with union of size at most $s$ and empty intersection. Answering a question of Frankl, we present some structural results for families that are $(d,s)$-conditionally intersecting with $s\ge 2k+d-3$, and families that are $(k,2k)$-conditionally intersecting. As applications of our structural results, we present some new proofs to the upper bounds for the size of the following $k$-uniform families on $[n]$. (a) $(d,2k+d-3)$-condi…

Connected-Intersecting Families of Graphs

For a graph property $\mathcal{P}$ and a common vertex set $V = \{1, 2, \ldots, n\}$, a family of graphs on $V$ is \emph{$\mathcal{P}$-intersecting} iff $G \cap H$ satisfies $\mathcal{P}$ for all $G,H$ in the family. Addressing a question of Chung, Graham, Frankl, and Shearer, we explore---for various $\mathcal{P}$---the maximum cardinality among all $\mathcal{P}$-intersecting families of graphs. In the connected-intersecting case, we resolve the question completely by a short linear algebraic proof showing this maximum is attained by taking all graphs containing a fixed spanning tree (though …

非均匀分布下的并闭集猜想

The union-closed sets conjecture for non-uniform distributions

本文将 Frankl 并闭集猜想推广到非均匀概率分布，证明无穷多种分布下的版本，并在交闭形式中由逆温度有正下界的 Maxwell–Boltzmann 分布推出结论。

Degree-doubling graph families

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.

有限向量空间中的极值子空间覆盖

Extremal subspace covers in finite vector spaces

本文研究有限向量空间中真子空间覆盖的极值问题，确定了 $m$ 个真子空间并集的最大大小并分类所有等式情形，同时引入基阻塞族并证明其最小大小为 $q$，给出统一极值组合描述。

关于并闭集猜想与 Reimer 平均集合大小定理的注记

Note on the union-closed sets conjecture and Reimer's average set size theorem

本文针对并闭集猜想与 Reimer 条件，提出了一种在成员集大小有任意固定下界时构造无穷多个不满足充裕条件反例的通用方法，并探讨了这些反例的相关性质及其与并闭集的差距。

The Linear Relaxation of an Integer Program for the Union-Closed Conjecture

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. Let $f(n,a)$ be the maximum 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 some $a,n\in \mathbb{N}^+$. Proving that $f(n,a)\leq 2a$ for all $a, n \in \mathbb{N}^+$ is equivalent to proving the Frankl conjecture. By considering the linear relaxation of the integer programming formulation that was proposed in New Conjectures for Union-Closed Famil…