Largest family without A union B contained in C intersect D

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 $q-2$.

An extension of the Erdős-Ko-Rado theorem to set-wise $2$-intersecting families of perfect matchings

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 $t\geq 2$.

非抵消相交猜想

The Non-Cancelling Intersections Conjecture

本文提出集族相交领域的非抵消相交猜想，给出其基于布尔格主下集的等价表述，并证明了相关部分结果，为集族测度与集合运算的关联研究提供了新的猜想方向。

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。

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。

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。

An Erdős-Ko-Rado Theorem for unions of length 2 paths

A family of sets is intersecting if any two sets in the family intersect. Given a graph $G$ and an integer $r\geq 1$, let $\mathcal{I}^{(r)}(G)$ denote the family of independent sets of size $r$ of $G$. For a vertex $v$ of $G$, the family of independent sets of size $r$ that contain $v$ is called an $r$-star. Then $G$ is said to be $r$-EKR if no intersecting subfamily of $ \mathcal{I}^{(r)}(G)$ is bigger than the largest $r$-star. Let $n$ be a positive integer, and let $G$ consist of the disjoint union of $n$ paths each of length 2. We prove that if $1 \leq r \leq n/2$, then $G$ is $r$-EKR. Th…

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.

对由几乎不相交的团并集构成的超图进行着色

Coloring hypergraphs that are the union of nearly disjoint cliques

本文研究由两两小交集团并集构成的超图最大色数，构造了团数量为多项式级的近似designs以证明其具有大色数，并通过随机贪婪超图过程对交集至多为1的情况给出下界，同时探讨了有限射影/仿影平面中最大caps数量的界。

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…

关于布尔多项式与并封闭猜想

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。

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 $\mathcal{H}$ …

On $t$-intersecting Hypergraphs with Minimum Positive Codegrees

For a hypergraph $\mathcal{H}$, define the minimum positive codegree $δ_i^+(\mathcal{H})$ to be the largest integer $k$ such that every $i$-set which is contained in at least one edge of $\mathcal{H}$ is contained in at least $k$ edges. For $1\le s\le k,t$ and $t\le r$, we prove that for $n$-vertex $t$-intersecting $r$-graphs $\mathcal{H}$ with $δ_{r-s}^+(\mathcal{H})>{k-1\choose s}$, the unique hypergraph with the maximum number of edges is the hypergraph $\mathcal{H}$ consisting of every edge which intersects a set of size $2k-2s+t$ in at least $k-s+t$ vertices provided $n$ is sufficiently l…

Maximum size intersecting families of bounded minimum positive co-degree

Let $\mathcal{H}$ be an $r$-uniform hypergraph. The \emph{minimum positive co-degree} of $\mathcal{H}$, denoted by $δ_{r-1}^+(\mathcal{H})$, is the minimum $k$ such that if $S$ is an $(r-1)$-set contained in a hyperedge of $\mathcal{H}$, then $S$ is contained in at least $k$ hyperedges of $\mathcal{H}$. For $r\geq k$ fixed and $n$ sufficiently large, we determine the maximum possible size of an intersecting $r$-uniform $n$-vertex hypergraph with minimum positive co-degree $δ_{r-1}^+(\mathcal{H}) \geq k$ and characterize the unique hypergraph attaining this maximum. This generalizes the Erd\H o…

A constant lower bound for the union-closed sets conjecture

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)…

并闭集猜想的一个改进下界

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。

Non-trivial $t$-intersecting families for symplectic polar spaces

Let $\mathscr{P}$ be a symplectic polar space over a finite field $\mathbb{F}_q$, and $\mathscr{P}_m$ denote the set of all $m$-dimensional subspaces in $\mathscr{P}$. We say a $t$-intersecting subfamily of $\mathscr{P}_m$ is trivial if there exists a $t$-dimensional subspace contained in each member of this family. In this paper, we determine the structure of maximum sized non-trivial $t$-intersecting subfamilies of $\mathscr{P}_m$.

用平面覆盖点集

Covering points with planes

针对向量空间中点集$S$的真子集被指定维度平面并集覆盖但$S$本身不被覆盖时$|S|$的最大规模问题，证明了一般上界（部分情况紧），构造反例并证明特殊情况上界，还研究了拟阵中的类似问题。

一类关于交闭集系的不等式

A class of inequalities for intersection-closed set systems

针对交闭集族中的 Frankl 猜想，本文证明一类不等式，并由其推出该猜想，为猜想提供了新的不等式依据。

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 …

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.

Critical and Maximum Independent Sets of a Graph

Let G be a simple graph with vertex set V(G). A subset S of V(G) is independent if no two vertices from S are adjacent. By Ind(G) we mean the family of all independent sets of G while core(G) and corona(G) denote the intersection and the union of all maximum independent sets, respectively. The number d(X)= |X|-|N(X)| is the difference of the set of vertices X, and an independent set A is critical if d(A)=max{d(I):I belongs to Ind(G)} (Zhang, 1990). Let ker(G) and diadem(G) be the intersection and union, respectively, of all critical independent sets of G (Levit and Mandrescu, 2012). In this pa…

Monotonic Properties of Collections of Maximum Independent Sets of a Graph

Let G be a simple graph with vertex set V(G). A subset S of V(G) is independent if no two vertices from S are adjacent. The graph G is known to be a Konig-Egervary if alpha(G) + mu(G)= |V(G)|, where alpha(G) denotes the size of a maximum independent set and mu(G) is the cardinality of a maximum matching. Let Omega(G) denote the family of all maximum independent sets, and f be the function from the set of subcollections Gamma of Omega(G) such that f(Gamma) = (the cardinality of the union of elements of Gamma) + (the cardinality of the intersection of elements of Gamma). Our main finding claims …

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…