Mauro Di Nasso
We consider shifts of a set $A\subseteq\mathbb{N}$ by elements from another set $B\subseteq\mathbb{N}$, and prove intersection properties according to the relative asymptotic size of $A$ and $B$. A consequence of our main theorem is the following: If $A=\{a_n\}$ is such that $a_n=o(n^{k/k-1})$, then the $k$-recurrence set $R_k(A)=\{x\mid |A\cap(A+x)|\ge k\}$ contains the distance sets of arbitrarily large finite sets.
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We isolate conditions on the relative size of sets of natural numbers $A,B$ that guarantee a nonempty intersection $Δ(A)\capΔ(B)\ne\emptyset$ of the corresponding sets of distances. Such conditions apply to a large class of zero density sets. We also show that a variant of Khintchine's Recurrence Theorem holds for all infinite sets $A=\{a_1<a_2<...\}$ with $a_n\ll n^{3/2}$.
The intersection of two linear sets of rank 3 in PG(1,q^3)
研究PG(1,q^3)中两个秩为3的线性集的交集，确定其所有可能交集大小，描述最大交集集结构并给出存在性结果。
In this article we construct new minimal intersection sets in ${\mathrm{AG}}(r,q^2)$ sporting three intersection numbers with hyperplanes; we then use these sets to obtain linear error correcting codes with few weights, whose weight enumerator we also determine. Furthermore, we provide a new family of three-character multisets in ${\mathrm{PG}}(r,q^2)$ with $r$ even and we also compute their weight distribution.
Inequivalence of skew Hadamard difference sets and triple intersection numbers modulo a prime
最近，Feng和Xiang \cite{FX113}在初等阿贝尔群中发现了一种新的斜Hadamard差集的构造。本文引入了一个新的斜Hadamard差集等价不变量，即模素数的三重相交数，并讨论了凤翔斜Hadamard差集与Paley差集的不等价性.因此，我们表明，他们的建设产生无穷多个斜Hadamard差集不等价于Paley差集。
The aim of this paper is to investigate the intersection problem between two linear sets in the projective line over a finite field. In particular, we analyze the intersection between two clubs with eventually different maximum fields of linearity. Also, we analyze the intersection between the linear set defined by the polynomial $αx^{q^k}+βx$ and other linear sets having the same rank; this family contains the linear set of pseudoregulus type defined by $x^q$. The strategy relies on the study of certain algebraic curves whose rational points describe the intersection of the two linear sets. A…
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 two-intersection set with parameters $(j;α,β)$ for a block design is a $j$-subset of the point set of the design, which intersects every block in $α$ or $β$ points. In this paper, we show the existence of a two-intersection set with parameters $(2m^2-m+1;m^2-m,m^2)$ for the block design obtained from translations of the set of nonzero squares in the finite field of order $q=4m^2+1$. As an application, we give a construction of conference matrices with maximum excess based on the two-intersection sets.
Intersection patterns of convex sets via simplicial complexes, a survey
本综述的任务是介绍关于凸集交模式的各种结果。单纯复形的观点是研究交叉模式的主要工具之一。我们回顾了$d$ -可表示、$d$ -可折叠和$d$ -Leray单纯复形的定义，这些定义对我们的研究非常有用.我们研究了这些概念之间的差异，我们也专注于计算复杂性识别它们。在综述中给出了一系列Helly型定理，并讨论了上述概念对这些定理的重要作用。我们还考虑交叉模式的良好覆盖，推广集合的凸集（集可能是“曲线”，但是，他们的交集不能太复杂）。我们主要关注新的结果。
A numerical monoid is an additive submonoid of the non-negative integers. Given a numerical monoid $S$, consider the family of "shifted" monoids $M_n$ obtained by adding $n$ to each generator of $S$. In this paper, we characterize the Apéry set of $M_n$ in terms of the Apéry set of the base monoid $S$ when $n$ is sufficiently large. We give a highly efficient algorithm for computing the Apéry set of $M_n$ in this case, and prove that several numerical monoid invariants, such as the genus and Frobenius number, are eventually quasipolynomial as a function of $n$.
Let $\mathcal A=\{A_1,\ldots,A_n\}$ be a family of sets in the plane. For $0 \leq i < n$, denote by $f_i$ the number of subsets $σ$ of $\{1,\ldots,n\}$ of cardinality $i+1$ that satisfy $\bigcap_{i \in σ} A_i \neq \emptyset$. Let $k \geq 2$ be an integer. We prove that if each $k$-wise and $(k+1)$-wise intersection of sets from $\mathcal A$ is empty, or a single point, or both open and path-connected, then $f_{k+1}=0$ implies $f_k \leq cf_{k-1}$ for some positive constant $c$ depending only on $k$. Similarly, let $b \geq 2, k > 2b$ be integers. We prove that if each $k$-wise or $(k+1)$-wise in…
Given sets $F_1, \ldots ,F_n$, a {\em partial rainbow function} is a partial choice function of the sets $F_i$. A {\em partial rainbow set} is the range of a partial rainbow function. Aharoni and Berger \cite{AhBer} conjectured that if $M$ and $N$ are matroids on the same ground set, and $F_1, \ldots ,F_n$ are pairwise disjoint sets of size $n$ belonging to $M \cap N$, then there exists a rainbow set of size $n-1$ belonging to $M \cap N$. Following an idea of Woolbright and Brower-de Vries-Wieringa, we prove that there exists such a rainbow set of size at least $n-\sqrt{n}$.
$s$-Shunt Intersection Graph of a Graph
本文首次提出并研究由给定图的 $s$-弧生成的 $s$-分流交图，拓展了交图理论的研究类型。
In this article, constant dimension subspace codes whose codewords have subspace distance in a prescribed set of integers, are considered. The easiest example of such an object is a {\it junta}; i.e. a subspace code in which all codewords go through a common subspace. We focus on the case when only two intersection values for the codewords, are assigned. In such a case we determine an upper bound for the dimension of the vector space spanned by the elements of a non-junta code. In addition, if the two intersection values are consecutive, we prove that such a bound is tight, and classify the ex…
A $(k,k-t)$-SCID (set of Subspaces with Constant Intersection Dimension) is a set of $k$-dimensional vector spaces that have pairwise intersections of dimension $k-t$. Let $\mathcal{C}=\{π_1,\ldots,π_n\}$ be a $(k,k-t)$-SCID. Define $S:=\langle π_1, \ldots, π_n \rangle$ and $I:=\langle π_i \cap π_j \mid 1 \leq i < j \leq n \rangle$. We establish several upper bounds for $\dim S + \dim I$ in different situations. We give a spectrum result for the case $(n-1)(k-t)\leq k$ and for the case $n\leq\frac{q^{t(n-η)}-1}{q^t-1}$, giving examples of $(k,k-t)$-SCIDs reaching a large interval of values for…
The Intersection of All Maximum Stable Sets of a Tree and its Pendant Vertices
Nemhauser和Trotter的一个定理保证了在一定的条件下，图G的一个独立集可以扩大到这个图的一个极大独立集.例如，任何仅由单纯顶点组成的独立集都包含在G的极大独立集中。在本文中，我们证明了一个逆断言是真实的树的顺序大于一，事实上，所有的单纯顶点是悬垂。也就是说，我们证明了这样的树的任何最大独立集至少包含一个悬垂顶点。证明了若T不具有完美匹配，则T的最大稳定集的交集中包含一个至少由两个悬垂顶点组成的独立集.对于树，上面的断言也是Hammer、Hansen和Simeone的一个结果的加强，指出如果G的一半阶小于G的最大独立集的基数，则其所有最大稳定集的交集是非空的。
We prove that for a connected simple graph $G$ with $n\le 10$ vertices, and two longest paths $C$ and $D$ in $G$, the intersection of vertex sets $V(C)\cap V(D)$ is a separator. This shows that the graph found previously with $n=11$, in which the complement of the intersection of vertex sets $V(C)\cap V(D)$ of two longest paths is connected, is minimal.
In this paper, we investigate the classes of matroid intersection admitting a solution for the problem of partitioning the ground set $E$ into $k$ common independent sets, where $E$ can be partitioned into $k$ independent sets in each of the two matroids. For this problem, we present a new approach building upon the generalized-polymatroid intersection theorem. We exhibit that this approach offers alternative proofs and unified understandings of previous results showing that the problem has a solution for the intersection of two laminar matroids and that of two matroids without $(k+1)$-spanned…
We derive sharp upper and lower bounds on the number of intersection points and closed regions that can occur in sets of line segments with certain structure, in terms of the number of segments. We consider sets of segments whose underlying planar graphs are Halin graphs, cactus graphs, maximal planar graphs, and triangle-free planar graphs, as well as randomly produced segment sets. We also apply these results to a variant of the Erdős-Faber-Lovász (EFL) Conjecture stating that the intersection points of $m$ segments can be colored with $m$ colors so that no segment contains points with the s…
Shifted and Shiftless Partition Identities II
设S和T是正整数的集合，a是固定的正整数。a移位的划分恒等式具有形式p（S，n）=p（T，n-a），对于所有大于或等于a的n。这里p（S，n）是n的部分是S的元素的划分数。对于所有已知的非平凡移位分拆恒等式，集合S和T是对某个M的模M的等差数列的并。1987年，Andrews发现了两个1-移位的例子（M=32，40），并询问是否还有更多的例子。1989年，卡尔瓦德又提出了6个。2000年，第一作者用计算机搜索发现了59个新的1-移位恒等式，并展示了如何用模函数理论证明这些恒等式。某些移位恒等式的模变换导致无移位的划分恒等式。再次设a是一个固定的正整数，S，T是不同的正整数集。无移位划分恒等式的形式为p（S，n）=p（T，n），其中n不等于a。在本文中，我们显示，除了在一种情况下，所有已知的1-移位和shiftless身份如下的四个参数θ函数身份由于雅可比。证明了新的移位和无移位划分恒等式。
A {\em string graph} is the intersection graph of a family of continuous arcs in the plane. The intersection graph of a family of plane convex sets is a string graph, but not all string graphs can be obtained in this way. We prove the following structure theorem conjectured by Janson and Uzzell: The vertex set of {\em almost all} string graphs on $n$ vertices can be partitioned into {\em five} cliques such that some pair of them is not connected by any edge ($n\rightarrow\infty$). We also show that every graph with the above property is an intersection graph of plane convex sets. As a corollar…
Cross-intersection theorems for uniform partitions of finite sets
本文研究有限集 c-均匀划分中的交叉 t-相交族，证明乘积型极值结果，并刻画规模和最大的族，由此得到均匀集合划分的 EKR 与 Hilton-Milner 型定理。
Elements represented as intersections of sets
该数学研究确定了[n]的表示族集的最小基数为Sperner函数的离散逆s(n)，并给出相关推广、应用及渐近结果。
The Intersection of All Maximum Stable Sets of a Tree and its Pendant Vertices
图G的稳定集是指图G中互不邻顶点的集合，α（G）是图G的最大独立集的大小，核（G）是图G的所有最大稳定集的交集.本文证明了在一棵树T中，任意一个大于1的独立集，其大小大于或等于n/2，至少包含一个悬挂点.因此，我们推出，任何最大独立集在一棵树包含至少一个悬垂顶点。我们的主要发现是定理声称，如果T不拥有完美的匹配，那么至少有两个悬垂顶点属于核心（T）。Levit和Mandrescu证明了如果G是阶至少为2的连通二部图，则核心（G）的大小不等于1，而我们的新陈述揭示了树的所有最大稳定集的交集的额外结构。上面的断言给出了Hammer，Hansen和Simeone的一个结果的精炼，声明如果图G的阶数小于2*alpha（G），则core（G）是非空的，并且还给出了Jamison，Gunter，Hartnel和Rall以及Zito的结果，声明对于阶数至少为2的树T，core（G）的大小不同于1。
A family of sets in the plane is simple if the intersection of its any subfamily is arc-connected, and it is pierced by a line $L$ if the intersection of its any member with $L$ is a nonempty segment. It is proved that the intersection graphs of simple families of compact arc-connected sets in the plane pierced by a common line have chromatic number bounded by a function of their clique number.
In this paper we give a classification of complete intersection vanishing ideals on parameterized sets of clutter type over finite fields.
Avoiding intersections of given size in finite affine spaces AG(n,2)
研究 AG(n,2) 中仿射子空间与点集的交集大小，部分确定了不可避免的局部密度，构造了基于子空间规避集的点集，并改进了其上界。
Finding Hamilton cycles in random intersection graphs
本文研究随机交集图G(n,m,p)中哈密顿回路的寻找问题，分析经典算法HAM的效率，证明其构造哈密顿回路的阈值函数与最小度至少为2的阈值函数相同，且HAM不依赖随机集合族的结构。
Following an approach presented by N. Frantzikinakis, B. Host and B. Kra, we show that the parameters in the multidimensional Szemerédi theorem for closest integer polynomials have non-empty intersection with the set of shifted primes $\mathbb{P}-1$ (or similarly of $\mathbb{P}+1$). Using the Furstenberg Correspondence Principle, we show this result by recasting it as a polynomial multiple recurrence result in measure ergodic theory. Furthermore, we obtain integer part polynomial convergence results by the same method, which is a transference principle that enables one to deduce results for $\…
Given a finite set $X$ of points in $R^n$ and a family $F$ of sets generated by the pairs of points of $X$, we determine volumetric and structural conditions for the sets that allow us to guarantee the existence of a positive-fraction subfamily $F'$ of $F$ for which the sets have non-empty intersection. This allows us to show the existence of weak epsilon-nets for these families. We also prove a topological variation of the existence of weak epsilon-nets for convex sets.
It is proved that there are triangle-free intersection graphs of line segments in the plane with arbitrarily small ratio between the maximum size of an independent set and the total number of vertices.
We introduce the cycle intersection graph of a graph, an adaptation of the cycle graph of a graph, and use the structure of these graphs to prove an upper bound for the decycling number of all even graphs. This bound is shown to be significantly better when an even graph admits a cycle decomposition in which any two cycles intersect in at most one vertex. Links between the cycle rank of the cycle intersection graph of an even graph and the decycling number of the even graph itself are found. The problem of choosing an ideal cycle decomposition is addressed and is presented as an optimization p…
Peres--Schlag's nonempty-interior problem and a shifted-product variant for product sets
研究 Peres--Schlag 非空内部问题在有限域和欧几里得空间中的乘积集变体，证明当集合大小或维数超过阈值时，线性组合或移位乘积覆盖整个空间或包含开区间。
A note on the multiplicative structure of an additively shifted product set, AA+1
我们考虑AA+1形式的集合的乘法结构，其中A是一个大的，有限的实数集。特别地，我们证明了加法移位的积集，AA+1必须有一个大的部分以外的任何广义等比数列的可比长度。我们在有限域中也证明了一个类似的结果。
Let $\mathcal F\subset 2^{[n]}$ be a family in which any three sets have non-empty intersection and any two sets have at least $38$ elements in common. The nearly best possible bound $|\mathcal F|\le 2^{n-2}$ is proved. We believe that $38$ can be replaced by $3$ and provide a simple-looking conjecture that would imply this.
Global Product Intersection Sets in Semigroups
本文研究了半群中乘积交集的性质，证明了当Q的大小大于等于2时，任何包含1的子集均可作为乘积交集出现，解决了Nathanson提出的问题。
The main result is a common generalization of results on lower bounds for the chromatic number of r-uniform hypergraphs and some of the major theorems in Tverberg-type theory, which is concerned with the intersection pattern of faces in a simplicial complex when continuously mapped to Euclidean space. As an application we get a simple proof of a generalization of a result of Kriz for certain parameters. This specializes to a short and simple proof of Kneser's conjecture. Moreover, combining this result with recent work of Mabillard and Wagner we show that the existence of certain equivariant m…
Ergodic averages and the large intersection property along IP sets
本文研究沿IP集的多个遍历平均，给出均值收敛与发散准则及收敛到不变投影的充分条件，证明有理障碍情形由幂零系统控制，并获逐点收敛及相关性的大交集性质。
We say that a family $\mathcal F$ of $k$-element sets is a {\it $j$-junta} if there is a set $J$ of size $j$ such that, for any $F$, its presence in $\mathcal F$ depends on its intersection with $J$ only. Approximating arbitrary families by $j$-juntas with small $j$ is a recent powerful technique in extremal set theory. The weak point of all known junta approximation results is that they work in the range $n>Ck$, where $C$ is an extremely fast growing function of the input parameters, such as the quality of approximation or the number of families we simultaneously approximate. We say that a fa…
Quantitative bounds for sets lacking polynomial progressions with shifted prime difference
该研究证明了移位参数限制在移位素数集合的多项式进展的定量Szemerédi型定理，改进了线性情形结果，不同次数非线性构型的定量界为首创，结合了整数多项式构型与素数Gowers一致性界等方法。
On the Intersection of All Critical Sets of a Unicyclic Graph
一个集合S在图G中是独立的，如果S中没有两个顶点是相邻的。独立数alpha（G）是最大独立集的基数，而mu（G）是G中最大匹配的大小。如果alpha（G）+mu（G）= |V|，则G=（V，E）称为Konig-Egervary图.数字d_{c}（G）=max{ |A| -|N（A）|}称为G的临界差（Zhang，1990）。通过core（G）（corona（G）），我们表示所有最大独立集的交集（并），而ker（G）表示所有临界独立集的交集。只有一个圈的连通图称为单圈图。已知ker（G）是每个图G的core（G）的子集，而等式对于二分图也成立（Levit and Mandrescu，2011）。对于Konig-Egervary单圈图，|核心（G）|-|克（G）|可以等于任何非负整数。本文证明了：如果G是一个非Konig-Egervary单圈图，则：（i）ker（G）= core（G）;|电晕（G）|+|核心（G）|=2*alpha（G）+1。注意|电晕（G）|+|核心（G）|=2*alpha（G）对每个Konig-Egervary图成立。
On Mattila's intersection problem in the plane: sharp dimension and positive measure of rotated difference sets
该论文针对平面上的Mattila相交问题，证明了旋转差集的Hausdorff维数下界的精确性及对应正测度结果，所用方法含角分解等，还探讨了其在相交模式上的应用。
We propose the concepts of intersection distribution and non-hitting index, which can be viewed from two related perspectives. The first one concerns a point set $S$ of size $q+1$ in the classical projective plane $PG(2,q)$, where the intersection distribution of $S$ indicates the intersection pattern between $S$ and the lines in $PG(2,q)$. The second one relates to a polynomial $f$ over a finite field $\mathbb{F}_q$, where the intersection distribution of $f$ records an overall distribution property of a collection of polynomials $\{f(x)+cx \mid c \in \mathbb{F}_q\}$. These two perspectives a…
Problems and results on intersections of product sets and sumsets in semigroups
研究半群中乘积集与和集交集的性质，探讨乘积交集集H(A_q)的结构与特性。
The cop number of a graph $G$ is the smallest $k$ such that $k$ cops win the game of cops and robber on $G$. We investigate the maximum cop number of geometric intersection graphs, which are graphs whose vertices are represented by geometric shapes and edges by their intersections. We establish the following dichotomy for previously studied classes of intersection graphs: The intersection graphs of arc-connected sets in the plane (called string graphs) have cop number at most 15, and more generally, the intersection graphs of arc-connected subsets of a surface have cop number at most $10g+15$ …
On regular sets of affine type in finite Desarguesian planes and related codes
本文研究有限Desarguesian平面中仿射型正则点集，提供一般构造方法与无限族，并据此获得具有特定权重的√q-可分码及其重量枚举子。
A Helly-type theorem for semi-monotone sets and monotone maps
我们考虑定义在实数上的O-极小结构上的集合和映射，例如实半代数或次解析集合。一个{ \em单调映射}是通常的一元单调函数的多维推广，而单调映射的图的闭包是紧凸集的推广.在一个常数相同的函数的特殊情况下，这样的图被称为{ \em半单调集}。单调映射的图通常是非凸的，它们的交集与凸集的交集不同，可以是拓扑复杂的。特别地，这样的交集不一定是单调映射的图。然而，我们证明了一个Helly型定理，即对于$\Real^n$的子集的有限族，如果基数至多为$n+1$的子族的所有交集都是非空的且是单调映射的图，则整个族的交集都是非空的且是单调映射的图。
Given a family of sets on the plane, we say that the family is intersecting if for any two sets from the family their interiors intersect. In this paper, we study intersecting families of triangles with vertices in a given set of points. In particular, we show that if a set $P$ of $n$ points is in convex position, then the largest intersecting family of triangles with vertices in $P$ contains at most $(\frac{1}{4}+o(1))\binom{n}{3}$ triangles.
When each vertex is assigned a set, the intersection graph generated by the sets is the graph in which two distinct vertices are joined by an edge if and only if their assigned sets have a nonempty intersection. An interval graph is an intersection graph generated by intervals in the real line. A chordal graph can be considered as an intersection graph generated by subtrees of a tree. In 1999, Karoński, Scheinerman and Singer-Cohen [Combin Probab Comput 8 (1999), 131--159] introduced a random intersection graph by taking randomly assigned sets. The random intersection graph $G(n,m;p)$ has $n$ …
On constrained intersection representations of graphs and digraphs
本文研究DAG最优有向交集表示问题，证明其在无三角形且哈密顿的DAG类中可多项式求解，并揭示与其他交集表示模型的关系。
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