Raffaele Scandone
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 independent samples from the uniform distribution over a union-closed family.
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Several results about the union-closed sets conjecture are presented.
In this very short note, we give a counterexample to a recent conjecture of Gilmer which would have implied the union-closed conjecture.
Proof of Union-Closed Sets Conjecture
该研究完成了并闭集猜想的证明工作。
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。
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 $\mathcal{A}$.
We show that the Union-Closed Conjecture holds for the union-closed family generated by the cyclic translates of any fixed set.
In a recent breakthrough, Gilmer proved the union closed conjecture up to a constant factor. Using Gilmer's method and additional ideas, Chase and Lovett proved an optimal result for almost union-closed set systems. Here that result is extended to higher order unions.
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 F(n, m), recorded as H(F). we prove that for any given union-closed family F, union-closed sets conjecture holds if its height number H(F) no more than 3 or no less than n-1. Since the height number H(F) is a positive integer which is bounded between 1 to n. As an attempted approach and framework, if we can prove union-closed sets conjecture holds for all possi…
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 conjectured that his technique can be pushed to the constant $\frac{3-\sqrt{5}}{2}$. We verify his conjecture; show that it extends to approximate union closed set systems, where for nearly all pairs of sets their union belong to the set system; and show that for such set systems this bound is optimal.
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.
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 family. In this paper, we introduce two stronger versions of Frankl's conjecture and give a partial proof. Three related questions are introduced.
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 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…
Frankl's union-closed sets conjecture states that in every finite union-closed set of sets, there is an element that is contained in at least half of the member-sets (provided there are at least two members). The conjecture has an equivalent formulation in terms of graphs: In every bipartite graph with least one edge, both colour classes contain a vertex belonging to at most half of the maximal stable sets. We prove that, for every fixed edge-probability, almost every random bipartite graph almost satisfies Frankl's conjecture.
The union-closed set conjecture is true
本文证明了 Peter Frankl 于 20 世纪 70 年代末提出的并闭集猜想：任何含非空集合的有限并闭族，都有元素至少出现在一半成员中。
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.
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.
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 $n$.
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}…
On the lattice formulation of the union-closed sets conjecture
本文研究并集闭集猜想的格结构，提出关于最小反例的必要条件。
On Boolean polynomials and the Union-Closed Conjecture
构造布尔多项式ICC_{m,n}(X)将并封闭猜想转化为多项式为零的条件，并利用等价形式交封闭猜想进行研究。
The weighted union-set conjecture is false
本文构造反例否证加权并集猜想，证明在满足特定权重条件下，每个元素可仅占极小权重比例。
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/…
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。
Better bounds for the union-closed sets conjecture using the entropy approach
本文结合 Sawin 的相关样本与 Gilmer 的熵方法，改进了并闭集猜想成立的已知常数界，并分析了熵方法的直觉与局限。
In 1979 Frankl conjectured that in a finite non-trivial union-closed collection 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 bipartiti…
A Study On The Graph Formulation Of Union Closed Sets Conjecture
研究并集封闭集猜想，通过连接集合与图的表述建立图论版本，证明相关定理，关注图的分解和悬挂顶点位置，将猜想有效性扩展到更广泛图结构类。
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.
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。
An improved lower bound for the union-closed set conjecture
本研究针对并闭集猜想的下界问题，改进Gilmer的方法并引入精确估计，将元素出现比例的下界从0.01提升至约0.38，还提出可进一步提升下界的思路，同时证否了Gilmer的一个相关猜想。
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…
Improved Lower Bound for Frankl's Union-Closed Sets Conjecture
本文通过验证 Gilmer 猜想的一个显式不等式，将 Frankl 并闭集合猜想中元素出现频率的下界改进到约 0.38，其中单变量不等式情形由计算机计算完成。
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.
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。
A Note on the Union-closed Sets Conjecture
本文研究了行在特定逻辑算子下封闭的非零二进制矩阵中1不少于0的列的存在性，并由此推导出Union-Closed Set Conjecture的一个较弱版本。
Supersaturation in union-closed families of sets
该数学研究证明了特定大小并族中k-链数量的极小化条件，明确了极小化族的唯一存在范围，为并族理论提供了关键结论。
We survey the state of the union-closed sets conjecture.
A 3-graph $\mathcal{F}$ is \emph{$U(s, 2s+1)$} if for any $s$ edges $e_1,...,e_s\in E(\mathcal{F})$, $|e_1\cup...\cup e_s|\leq 2s+1$. Frankl and Kupavskii (2020) proposed the following conjecture: For any $3$-graph $\mathcal{F}$ with $n$ vertices, if $\mathcal{F}$ is $U(s, 2s+1)$, then $$e(\mathcal{F})\leq \max\left\{{n-1\choose 2}, (n-s-1){s+1\choose 2}+{s+1\choose 3}, {2s+1\choose 3}\right\}.$$ In this paper, we confirm Frankl and Kupavskii's conjecture.
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…
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。
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。
We show that for any union-closed family $\mathcal{F} \subseteq 2^{[n]}, \mathcal{F} \neq \{\emptyset\}$, there exists an $i \in [n]$ which is contained in a $0.01$ fraction of the sets in $\mathcal{F}$. This is the first known constant lower bound, and improves upon the $Ω(\log_2(|\mathcal{F}|)^{-1})$ bounds of Knill and Wójick. Our result follows from an information theoretic strengthening of the conjecture. Specifically, we show that if $A, B$ are independent samples from a distribution over subsets of $[n]$ such that $Pr[i \in A] < 0.01$ for all $i$ and $H(A) > 0$, then $H(A \cup B) > H(A)…
Frequent elements in union-closed set families
本文结合 Gilmer 的熵方法与 Knill 的组合论证，证明了 Nagel 提出的关于并闭集族中第 k 个最常见元素出现频率的推广猜想，并刻画了取等集族及无穷极限下的渐近下界。
Note on the union-closed sets conjecture and Reimer's average set size theorem
本文针对并闭集猜想与 Reimer 条件，提出了一种在成员集大小有任意固定下界时构造无穷多个不满足充裕条件反例的通用方法，并探讨了这些反例的相关性质及其与并闭集的差距。
The following conjecture was proposed in 2010 by S. Lando. Let M and N be two unions of the same number of disjoint circles in a sphere. Then there exist two spheres in 3-space whose intersection is transversal and is a union of disjoint circles that is situated as M in one sphere and as N in the other . Define union A of disjoint circles to be situated in one sphere as union B of disjoint circles in the other sphere if there is a homeomorphism between these two spheres which maps A to B. In this paper we prove that there exists pair of sets of 7 circles in sphere, that is a counterexample to …
Local Configurations in Union-Closed Families
本文研究了并集闭集猜想，求得了最小整数$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。
A graceful difference labeling (gdl for short) of a directed graph G with vertex set V is a bijection f between V and {1,...,|V|} such that, when each arc uv is assigned the difference label f(v)-f(u), the resulting arc labels are distinct. We conjecture that all disjoint unions of circuits have a gdl, except in two particular cases. We prove partial results which support this conjecture.
Kuratowski proved that a finite graph embeds in the plane if it does not contain a subdivision of either K_5 or K_{3,3}, called Kuratowski subgraphs. A conjectured generalization of this result to all nonorientable surfaces says that a finite minimal forbidden subgraph for the nonorientable surface of genus g can be written as the union of g+1 Kuratowski subgraphs such that the union of each pair of these fails to embed in the projective plane, the union of each triple of these fails to embed in the Klein bottle if g >= 2, and the union of each triple of these fails to embed in the torus if g …
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。
Borodin & Kostochka conjectured that if maximum degree of a graph is greater than or equal to 9, then the chromatic number of the graph is less than or equal to maximum of ω and maximum degree minus 1. Here we prove that this Conjecture is true for {P3 UNION K1}-free graphs and {K2 UNION complement of K2}-free graphs.
Progress on the union-closed conjecture and offsprings in winter 2022-2023
本文综述了2022-2023年冬季并闭猜想及其变体的研究进展，重点介绍了Justin Gilmer首次证明并闭族中最常见元素比例存在常数下界的突破性贡献。
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…
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。
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…
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 $\…
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…
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。
If X is an n-element set, we call a family G of subsets of X a k-generator for X if every subset of X can be expressed as a union of at most k disjoint sets in G. Frein, Leveque and Sebo conjectured that for n > 2k, the smallest k-generators for X are obtained by taking a partition of X into classes of sizes as equal as possible, and taking the union of the power-sets of the classes. We prove this conjecture for all sufficiently large n when k = 2, and for n a sufficiently large multiple of k when k > 2.
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。
Beyond Liu's 0.382709 threshold for the union-closed sets conjecture
本文在 Liu 的框架内引入不同协议，结合独立协议与 Liu 的例 5，证明了并封闭集猜想的下界 $c_{\mathrm{UC}\ge c_{\mathrm L}$，并通过参数优化进一步将下界提升至 $0.3828852549667978$。
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