Imre Barany
We prove that given a finite collection of cylinders in $\R^3$ with the property that any two them intersect, then there is a line intersecting an $α$ fraction of the cylinders where $α=\frac 1{28}$. This is a special case of an interesting conjecture.
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In this paper, we analyze the structure of maximal sets of $k$-dimensional spaces in $\mathrm{PG}(n,q)$ pairwise intersecting in at least a $(k-2)$-dimensional space, for $3 \leq k\leq n-2$. We give an overview of the largest examples of these sets with size more than $f(k,q)=\max\{3q^4+6q^3+5q^2+q+1,θ_{k+1}+q^4+2q^3+3q^2\}$.
A $t$-intersecting constant dimension subspace code $C$ is a set of $k$-dimensional subspaces in a projective space PG(n,q), where distinct subspaces intersect in a $t$-dimensional subspace. A classical example of such a code is the sunflower, where all subspaces pass through the same $t$-space. The sunflower bound states that such a code is a sunflower if $|C| > \left( \frac {q^{k + 1} - q^{t + 1}}{q - 1} \right)^2 + \left( \frac {q^{k + 1} - q^{t + 1}}{q - 1} \right) + 1$. In this article we will look at the case $t=0$ and we will improve this bound for $q\geq 9$: a set $\mathcal{S}$ of $k$-…
A note on lenses in arrangements of pairwise intersecting circles in the plane
该研究证明平面上$n$个两两相交圆构成的族诱导的构形中透镜数至多为$2n-2$且界紧，还证明圆不相切时对应圆心几何图不含回避边对，其边数同样满足该上界。
On the number of digons in arrangements of pairwise intersecting circles
本文证明了 Grünbaum 于1972年提出的长期猜想，即平面上任意由 $n$ 个两两相交圆构成的简单排列中最多有 $2n-2$ 个二角形。
We show that the maximum number of pairwise intersecting positive homothets of a $d$-dimensional centrally symmetric convex body, none of which contains the center of another in its interior, is at most $3^{d+1}$. Also, we improve upper bounds for cardinalities of $k$-distance sets in Minkowski spaces.
On the number of pairwise touching cylinders in $\mathbb{R}^d$
研究在更高维度空间中两两相切圆柱体的最大数量，提出指数级上界并基于多项式等式和不等式的方法进行分析。
Complete Characterization on Maximum Pairwise Cross Intersecting Families (I)
本文研究了最大成对交叉相交族的结构，解决了Shi-Frankl-Qian提出的问题，并确定了对于所有$n\geq k_1+k_2$，$M(n, k_1, \dots, k_t)$的值。
The number of k-intersections of an intersecting family of r-sets
Erdos-Ko-Rado定理告诉我们从一个n-集合中的r-集合的相交族可以有多大，而Lovasz和Tuza的结果给出了从这样一个族中的集合的成对相交可以出现的单例数的界限。我们认为这些问题的一个自然的概括。给定一个n-集的r-集的相交族和1个\leq k \leq r，有多少个k-集可以作为该族中的集的两两相交出现？对于k=r和k=1，这简化为上述问题。当n足够大时，我们对k和r的所有值都能准确地回答这个问题。我们还刻画了极值族。
A convex polygon $A$ is related to a convex $m$-gon $K= \bigcap_{i=1}^m k_i^+$, where $k_1^+,..., k_m^+$ are the $m$ halfplanes whose intersection is equal to $K$, if $A$ is the intersection of halfplanes $a_1^+,...,a_l$, each of which is a translate of one of the $k_i^+$-s. The planar family ${\cal A}$ is related to $K$ if each $A \in {\cal A}$ is related to $K$. We prove that any family of pairwise intersecting convex sets related to a given $n$-gon has a finite piercing number which depends on $n$. In the general case we show $O(3^{n^3})$, while for a certain class of families, we decrease …
Mixed pairwise cross intersecting families (I)
本文首次研究混合型成对交叉相交族系统，在条件 $k_1+k_3\leq n < k_1+k_2$ 下确定了各族大小之和的最大值并刻画了所有极值结构，为最一般条件 $n\ge k_1+k_t$ 的解决奠定基础。
Non-uniform pairwise cross $t$-intersecting families
研究非均匀 t-相交族的上界，通过生成集方法和推拉方法证明了多个族的交叉 t-相交条件下的最大规模限制，并扩展了 Katona 和 Frankl 等人的结果。
Three intersection theorems are proved. First, we determine the size of the largest set system, where the system of the pairwise unions is l-intersecting. Then we investigate set systems where the union of any s sets intersect the union of any t sets. The maximal size of such a set system is determined exactly if s+t<5, and asymptotically if s+t>4. Finally, we exactly determine the maximal size of a k-uniform set system that has the above described (s,t)-union-intersecting property, for large enough n.
The maximum sum of sizes of non-empty cross $L$-intersecting families
本文研究非空交叉 $L$-交族的总大小上界，确定两族情形的最大值并刻画极值结构，且在充分大 $n$ 等条件下推广到同时满足两类交叉交条件的 $r$ 个族。
Non-empty pairwise cross-intersecting families
本文研究了非空两两交叉相交族，给出了其加权和的紧上界及达到上界的族结构刻画，统一了前人结果并扩展了适用范围，证明中利用了Kruskal-Katona定理并验证了关键函数的单峰性。
On Lines Crossing Pairwise Intersecting Convex Sets in Three Dimensions
研究证明在三维空间中，成对相交的凸集家族中存在一条直线可穿透Θ(n)个元素，解决了该问题的变种。
At most nine lines in Euclidean three-space have pairwise distance one
通过纯数学证明，在三维欧几里得空间中，最多存在九条直线使得任意两条之间的距离恰好为1，改进了此前部分依赖计算机的十一线结果。
A pseudocircle is a simple closed curve on the sphere or in the plane. The study of arrangements of pseudocircles was initiated by Grünbaum, who defined them as collections of simple closed curves that pairwise intersect in exactly two crossings. Grünbaum conjectured that the number of triangular cells $p_3$ in digon-free arrangements of $n$ pairwise intersecting pseudocircles is at least $2n-4$. We present examples to disprove this conjecture. With a recursive construction based on an example with $12$ pseudocircles and $16$ triangles we obtain a family with $p_3(\mathcal{A})/n \to 16/11 = 1.…
A long standing conjecture of Richter and Thomassen states that the total number of intersection points between any $n$ simple closed Jordan curves in the plane, so that any pair of them intersect and no three curves pass through the same point, is at least $(1-o(1))n^2$. We confirm the above conjecture in several important cases, including the case (1) when all curves are convex, and (2) when the family of curves can be partitioned into two equal classes such that each curve from the first class is touching every curve from the second class. (Two curves are said to be touching if they have pr…
In this paper we investigate Erdős-Ko-Rado theorems in ovoidal circle geometries. We prove that in Möbius planes of even order greater than 2, and ovoidal Laguerre planes of odd order, the largest families of circles which pairwise intersect in at least one point, consist of all circles through a fixed point. In ovoidal Laguerre planes of even order, a similar result holds, but there is one other type of largest family of pairwise intersecting circles. As a corollary, we prove that the largest families of polynomials over $\mathbb F_q$ of degree at most $k$, with $2 \leq k < q$, which pairwise…
Let $S$ be an $n$-punctured sphere, with $n \geq 3$. We prove that $\binom{n}{3}$ is the maximum size of a family of pairwise non-homotopic simple arcs on $S$ joining a fixed pair of distinct punctures of $S$ and pairwise intersecting at most twice. On the way, we show that a square annular diagram $A$ has a corner on each of its boundary paths if $A$ contains at least one square and the dual curves of $A$ are simple arcs joining the boundary paths of $A$ and pairwise intersecting at most once.
Anti-Ramsey Number of Intersecting Odd Cycles
本文确定了相交奇圈（由三角形和长度至少5的奇圈共享一个公共顶点构成）的anti-Ramsey数的精确值，推广了Liu等人关于仅含三角形情形的结果，适用于所有足够大的$n$。
Pairwise Intersection Profiles of Intersecting Families
本研究探讨相交族的成对相交分布，证明了在特定n的取值下，相交族的m阶相交有序对数量以完全星族为最大，且等号仅当族为置换下的星族时成立，还推导了相关泛函的最大值结论及稳定性结果。
A computer-assisted proof of Kuperberg's six-cylinder conjecture
研究库珀伯格六圆柱猜想，通过计算机辅助将其简化为有限案例分析，对2,954,984种情况验证有理系数多项式不等式，证明最多六个两两不重叠的无限单位圆柱可同时与单位球相切。
Partial covering arrays and a generalized Erdos-Ko-Rado property
经典的Erd \H os-Ko-Rado定理指出，如果$k\le\floor{n/2}$，则$[n]=\{0,1,...,n\}$的最大的成对相交$k$ -子集族的大小为${{n-1}\choose{k-1}}$。满足这种两两相交性质的$k$子集族称为EKR族。我们推广的EKR属性，并提供渐近下界的最大家庭${\cal A}$的$k$ -子集的$[n]$，满足以下属性的大小：对于每个$A,B,C\in{\cal A}$，每个四个集合$A\cap B\cap C;A\cap B\cap C^C; A\cap B^C\cap C; A^C\cap B\cap C$是非空的。这种广义EKR（GEKR）属性的动机，建议推广，并与固定重量3覆盖阵列的比较。我们的技术是概率性的。
We show that any set of distinct homotopy classes of simple closed curves on the torus that pairwise intersect at most $k$ times has size $k + O(\sqrt{k} \log k)$. Prior to this work, a lemma of Agol, together with the state of the art bounds for the size of prime gaps, implied the error term $O(k^{21/40})$, and in fact the assumption of the Riemann hypothesis improved this error term to the one we obtain $O(\sqrt{k} \log k)$. By contrast, our methods are elementary, combinatorial, and geometric.
Let $S^r(n)$ be the $r$-graph on $n$ vertices with parts $A$ and $B$, where the edges consist of all $r$-tuples with $1$ vertex in $A$ and $r-1$ vertices in $B$, and the sizes of $A$ and $B$ are chosen to maximise the number of edges. Let $M_t^r$ be the $r$-graph with $t$ pairwise disjoint edges. Given an $r$-graph $F$ and a positive integer $p\geq |V(F)|$, we define the {\em extension} of $F$, denoted by $H_{p}^{F}$ as follows: Label the vertices of $F$ as $v_1,\dots,v_{|V(F)|}$. Add new vertices $v_{|V(F)|+1},\dots,v_{p}$. For each pair of vertices $v_i,v_j, 1\le i<j \le p$ not contained in …
A starting point in the investigation of intersecting systems of subsets of a finite set is the elementary observation that the size of a family of pairwise intersecting subsets of a finite set [n]={1,...,n}, denoted by 2^{[n]}, is at most 2^{n-1}, with one of the extremal structures being the family comprised of all subsets of [n] containing a fixed element, called as a star. A longstanding conjecture of Chvátal aims to generalize this simple observation for all downsets of 2^{[n]}. In this note, we prove this conjecture for all downsets where every subset contains at most 3 elements.
The maximum number of digons formed by pairwise crossing pseudocircles
本研究将Grünbaum关于两两相交伪圆排列中二角形最大数量为2n−2的猜想，从柱面排列、几何圆情形推广至任意简单两两相交伪圆排列，给出了该猜想的完整证明，解决了悬置五十余年的公开问题。
The matching number of a family of subsets of an $n$-element set is the maximum number of pairwise disjoint sets. The families with matching number $1$ are called intersecting. The famous Erd\H os-Ko-Rado theorem determines the size of the largest intersecting family of $k$-sets. Its generalization to the families with larger matching numbers, known under the name of the Erdős Matching Conjecture, is still open for a wide range of parameters. In this paper, we address the degree versions of both theorems. More precisely, we give degree and $t$-degree versions of the Erdős-Ko-Rado and the Hilto…
An Erdős-Ko-Rado set in a block design is a set of pairwise intersecting blocks. In this article we study Erdős-Ko-Rado sets in 2-(v,k,1) designs, Steiner systems. The Steiner triple systems and other special classes are treated separately. For unitals we also determine an upper bound on the size of the second-largest maximal Erdos-Ko-Rado sets.
Erdős-Ko-Rado theorem and Hilton-Milner type theorem for $k$-partitions
本文研究$k$-划分的$t$-相交族，证明了在$n$足够大时最大平凡族由固定单元素块构成，并确定了最大非平凡族的结构，同时给出稳定性结果。
A monotone cylindrical graph is a topological graph drawn on an open cylinder with an infinite vertical axis satisfying the condition that every vertical line intersects every edge at most once. It is called simple if any pair of its edges have at most one point in common: an endpoint or a point at which they properly cross. We say that two edges are disjoint if they do not intersect. We show that every simple complete monotone cylindrical graph on $n$ vertices contains $Ω(n^{1-ε})$ pairwise disjoint edges for any $ε>0$. As a consequence, we show that every simple complete topological graph (d…
We prove that the Herzog-Schönheim Conjecture holds for any group $G$ of order smaller than $1440$. In other words we show that in any non-trivial coset partition $\{g_i U_i\}_{i=1}^n $ of $G$ there exist distinct $1 \leq i, j \leq n$ such that $[G:U_i]=[G:U_j]$. We also study interaction between the indices of subgroups having cosets with pairwise trivial intersection and harmonic integers. We prove that if $U_1$,...,$U_n$ are subgroups of $G$ which have pairwise trivially intersecting cosets and $n \leq 4$ then $[G:U_1]$,...,$[G:U_n]$ are harmonic integers.
Orientation of convex sets
本文提出了一种两两相交平面凸集族三元组定向的新定义，将其与满足内部性条件的偏3序（P3O）和全3序（T3O）进行比较，揭示了凸集定向与点集序型的本质差异，并探讨了满足(4,3)性质时的特性。
On the existence of factors intersecting sets of cycles in regular graphs
研究正则图中是否存在与给定循环集相交的因子，证明了比值t/r≥1/3的必要条件，并在特定情况下验证了其充分性。
Erdős-Ko-Rado Theorems for Paths in Graphs
本文研究图中长度-r路径族的1-相交和s-相交性质，证明太阳图和theta图满足EKR性质，并给出Hilton-Milner型结果和横贯数结论。
A minor-model of a graph $H$ in a graph $G$ is a subgraph of $G$ that can be contracted to $H$. We prove that for a positive integer $\ell$ and a non-empty planar graph $H$ with at least $\ell-1$ connected components, there exists a function $f_{H, \ell}:\mathbb{N}\rightarrow \mathbb{R}$ satisfying the property that every graph $G$ with a family of vertex subsets $Z_1, \ldots, Z_m$ contains either $k$ pairwise vertex-disjoint minor-models of $H$ each intersecting at least $\ell$ sets among prescribed vertex sets, or a vertex subset of size at most $f_{H, \ell}(k)$ that meets all such minor-mod…
A family of homothets of an o-symmetric convex body K in d-dimensional Euclidean space is called a Minkowski arrangement if no homothet contains the center of any other homothet in its interior. We show that any pairwise intersecting Minkowski arrangement of a d-dimensional convex body has at most $2\cdot 3^d$ members. This improves a result of Polyanskii (arXiv:1610.04400). Using similar ideas, we also give a proof the following result of Polyanskii: Let $K_1,\dots,K_n$ be a sequence of homothets of the o-symmetric convex body $K$, such that for any $i<j$, the center of $K_j$ lies on the boun…
On the VC-dimension of half-spaces with respect to convex sets
本文研究平面凸集关于半空间的VC维数，证明两两不相交凸集VC维数上界为3且紧，而一般凸集无界，线段情形VC维数至多5且紧，并给出几何覆盖与范围查询应用。
Probabilistic Extensions of the Erd\H os-Ko-Rado Property
经典的Erd \H os-Ko-Rado（EKR）定理指出，如果我们从一个固定大小的集合（n（n > 2k））中选择一个子集族，每个子集的大小为（k），那么最大可能的成对相交族的大小为（t ={n-1 \choose k-1}）。本文考虑随机选择的大小为t=t_n的族具有EKR性质（两两非空交）的概率，即$n$和$k=k_n$趋于无穷大，后者以特定的速率趋于无穷大.当$t$变大时，EKR性质不太可能发生，而当$t$变小时，EKR性质以高概率满足。我们使用Janson不等式推导出$t$的阈值。使用斯坦-陈方法，我们表明，分布的$X_0$，定义为不相交的子集对在我们的家庭，可以近似的泊松分布。我们扩展我们的结果，以产生类似的结论$X_i$，重叠的子集对的数量正好在$i$元素。最后，我们表明，联合分布$(X_0, X_1, ..., X_b)$可以近似为一个多维泊松向量与独立的组件。
Transversal generalizations of hyperplane equipartitions
该研究拓展了Dolnikov超平面横截定理，通过建立拓扑Radon型相交定理结合Gale对偶，证明满足更一般非不交条件的紧凸集族有多超平面公共横截，推广了双超平面下Grünbaum–Hadwiger–Ramos测度等分问题的最优结果，并给出单超平面下Dolnikov定理的新证明。
From arcs to curves: quadratic growth of 1-systems
本文证明可定向曲面上两两相交至多一次的简单闭曲线族数量至多为二次增长，引入郁金香、花朵和茎系统概念，解决了三十年前的问题。
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