Imre Bárány, Gil Kalai
In this paper, we present a variety of problems in the interface between combinatorics and geometry around the theorems of Helly, Radon, Carathéodory, and Tverberg. Through these problems we describe the fascinating area of Helly-type theorems, and explain some of its main themes and goals.
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Helly numbers for Quantitative Helly-type results
本文获得了三个 Helly 型结果，包括关于凸体族交集直径最优 Helly 数为 2d 的定量彩色 Helly 型定理，以及对数凹函数逐点最小值最优 Helly 数为 2d+1 的定理及其 Helly 数为 3d+1 的彩色版本。
Helly-type theorems for separated $d$-intervals
本文针对分离 $d$-区间导出的凸性空间，建立了包含 Radon 数、Helly 数、彩色及分数 Helly 定理和 $(p,q)$ 定理在内的系列 Helly 型定理，主要利用单纯复形和可坍缩性进行证明。
Helly-type theorems for monotone properties of boxes
提出统一方法证明盒子单调性质的Helly型定理，涵盖大体积或含点等性质，推广到H-凸集，并给出早期结果的新证明。
Helly type problems in convexity spaces
该研究报告了有界Radon数凸性空间组合性质的最新进展，讨论了Radon数与多种Helly相关性质、弱ε-网及(p,q)定理的关系，并应用该理论引入新一致超图类且证明其χ-有界。
Let X be a simplicial complex on the vertex set V. The rational Leray number L(X) of X is the minimal d such that the rational reduced homology of any induced subcomplex of X vanishes in dimensions d and above. Let πbe a simplicial map from X to a simplex Y, such that the cardinality of the preimage of any point in |Y| is at most r. It is shown that L(π(X)) \leq r L(X)+r-1. One consequence is a topological extension of a Helly type result of Amenta.
Quantitative Helly-type Theorems via Hypergraph Chains
本文提出一种组合框架分析定量 Helly 型问题，并据此证明了分数 Helly 数为 3d 的定量分数 Helly 定理及现有定量 Helly 定理的稳定性版本。
A Helly-Type Theorem for two-component convex sets
该研究针对任意固定维数下的双分量凸集族，证明了保证整体交集仍为双分量凸集的充分条件，解答了Gil Kalai的相关问题。
Let $H$ be a complete $r$-uniform hypergraph such that two vertices are marked in each edge as its `boundary' vertices. A linear ordering of the vertex set of $H$ is called an {\em agreeing linear order}, provided all vertices of each edge of $H$ lie between its two boundary vertices. We prove the following Helly-type theorem: if there is an {agreeing linear order} on the vertex set of every subhypergraph of $H$ with at most $2r-2$ vertices, then there is an agreeing linear order on the vertex set of $H$. We also show that the constant $2r-2$ cannot be reduced in the theorem. The case $r=3$ of…
Research on Helly-type theorems in combinatorial convex geometry has produced volumetric versions of Helly's theorem using witness sets and quantitative extensions of Doignon's theorem. This paper combines these philosophies and presents quantitative Helly-type theorems for the integer lattice with axis-parallel boxes as witness sets. Our main result shows that, while quantitative Helly numbers for the integer lattice grow polynomially in each fixed dimension, their variants with boxes as witness sets are uniformly bounded. We prove several colorful and fractional variations on this theorem. W…
We study versions of Helly's theorem that guarantee that the intersection of a family of convex sets in $R^d$ has a large diameter. This includes colourful, fractional and $(p,q)$ versions of Helly's theorem. In particular, the fractional and $(p,q)$ versions work with conditions where the corresponding Helly theorem does not. We also include variants of Tverberg's theorem, Bárány's point selection theorem and the existence of weak epsilon-nets for convex sets with diameter estimates.
A Helly-type theorem for diameter provides a bound on the diameter of the intersection of a finite family of convex sets in $\mathbb{R}^d$ given some information on the diameter of the intersection of all sufficiently small subfamilies. We prove fractional and colorful versions of a longstanding conjecture by Bárány, Katchalski, and Pach. We also show that a Minkowski norm admits an exact Helly-type theorem for diameter if and only if its unit ball is a polytope and prove a colorful version for those that do. Finally, we prove Helly-type theorems for the property of ``containing $k$ colinear i…
How many monochromatic paths, cycles or general trees does one need to cover all vertices of a given $r$-edge-coloured graph $G$? These problems were introduced in the 1960s and were intensively studied by various researchers over the last 50 years. In this paper, we establish a connection between this problem and the following natural Helly-type question in hypergraphs. Roughly speaking, this question asks for the maximum number of vertices needed to cover all the edges of a hypergraph $H$ if it is known that any collection of a few edges of $H$ has a small cover. We obtain quite accurate bou…
To any simple graph \(G\), the clique graph operator \(K\) assigns the graph \(K(G)\) which is the intersection graph of the maximal complete subgraphs of \(G\). The iterated clique graphs are defined by \(K^{0}(G)=G\) and \(K^{n}(G)=K(K^{n-1}(G))\) for \(n\geq 1\). We associate topological concepts to graphs by means of the simplicial complex \(\mathrm{Cl}(G)\) of complete subgraphs of \(G\). Hence we say that the graphs \(G_{1}\) and \(G_{2}\) are homotopic whenever \(\mathrm{Cl}(G_{1})\) and \(\mathrm{Cl}(G_{2})\) are. A graph \(G\) such that \(K^{n}(G)\simeq G\) for all \(n\geq1\) is calle…
Colorful fractional Helly theorem via weak saturation
本文通过组合归约将 fractional Helly theorem 及其 colorful 版本与极值组合学中的 weak saturation 问题相联系，利用基本线性代数论证，在不使用 exterior algebra 的情况下给出了最优界的新证明。
A hollow axis-aligned box is the boundary of the cartesian product of $d$ compact intervals in R^d. We show that for d\geq 3, if any 2^d of a collection of hollow axis-aligned boxes have non-empty intersection, then the whole collection has non-empty intersection; and if any 5 of a collection of hollow axis-aligned rectangles in R^2 have non-empty intersection, then the whole collection has non-empty intersection. The values 2^d for d\geq 3 and 5 for d=2 are the best possible in general. We also characterize the collections of hollow boxes which would be counterexamples if 2^d were lowered to …
Helly-type problems from a topological perspective
本文从拓扑学角度探讨了Helly型问题的最新进展，介绍了基于神经引理和非嵌入性的两种证明方法。
We introduce a new variant of quantitative Helly-type theorems: the minimal \emph{"homothetic distance"} of the intersection of a family of convex sets to the intersection of a subfamily of a fixed size. As an application, we establish the following quantitative Helly-type result for the \emph{diameter}. If $K$ is the intersection of finitely many convex bodies in $\mathbb{R}^d$, then one can select $2d$ of these bodies whose intersection is of diameter at most $(2d)^3\mathrm{diam}(K)$. The best previously known estimate, due to Brazitikos, is $c d^{11/2}$. Moreover, we confirm that the multip…
Helly Theorems for Generalized Turán Problems
本文提出一个广义Turán数的定理，通过树和图族的关系，证明了在特定条件下，Turán数要么具有Ω(n^{k+1})下界，要么具有O(ex(n, F)^k)上界，并引入了新的树Helly定理变体。
Helly groups
本文定义并系统研究在 Helly 图上几何作用的 Helly 群，给出丰富群类例子并证明其双自动性、分类空间、渐近锥、边界及两大猜想等性质。
Stability for Helly-type and triangle-free families
该文研究k-图中Helly型族与无三角形族的稳定性，在更宽松的n>2k条件下证明了非星型Helly族的大小上界，并推导了无三角形族对应上界的成立与不成立的参数范围。
This survey presents recent Helly-type geometric theorems published since the appearance of the last comprehensive survey, more than ten years ago. We discuss how such theorems continue to be influential in computational geometry and in optimization.
We consider quantitative versions of Helly-type questions, that is, instead of finding a point in the intersection, we bound the volume of the intersection. Our first main geometric result is a quantitative version of the Fractional Helly Theorem of Katchalski and Liu, the second one is a quantitative version of the $(p,q)$-Theorem of Alon and Kleitman.
The Helly number of Hamming balls and related problems
本文证明了半径受限的Hamming球的Helly定理变体，指出任意至多$2^{t+1}$个球子族有公共点即可保证全族有公共点，该界紧致且不依赖于环境空间维数。
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 $q-2$.
We prove discrete Helly-type theorems for pseudohalfplanes, which extend recent results of Jensen, Joshi and Ray about halfplanes. Among others we show that given a family of pseudohalfplanes $\cal H$ and a set of points $P$, if every triple of pseudohalfplanes has a common point in $P$ then there exists a set of at most two points that hits every pseudohalfplane of $\cal H$. We also prove that if every triple of points of $P$ is contained in a pseudohalfplane of $\cal H$ then there are two pseudohalfplanes of $\cal H$ that cover all points of $P$. To prove our results we regard pseudohalfplan…
Colorful positive bases decomposition and Helly-type results for cones
本文提出了一种彩色正基分解方法，证明了在特定条件下锥的Helly型结果，展示了该方法在优化问题中的应用。
The isomorphism problem is known to be efficiently solvable for interval graphs, while for the larger class of circular-arc graphs its complexity status stays open. We consider the intermediate class of intersection graphs for families of circular arcs that satisfy the Helly property. We solve the isomorphism problem for this class in logarithmic space. If an input graph has a Helly circular-arc model, our algorithm constructs it canonically, which means that the models constructed for isomorphic graphs are equal.
We prove that if any $\lfloor3d/2 \rfloor$ or fewer elements of a finite family of linear operators $\mathbb K^d\to \mathbb K^d$ ($\mathbb K$ is an arbitrary field) have a common eigenvector then all operators in the family have a common eigenvector. Moreover, $\lfloor 3d/2\rfloor$ cannot be replaced by a smaller number. Also, we study the following problem, achieving partial results: prove that if any $l=O(d)$ or fewer elements of a finite family of linear operators $\mathbb K^d\to \mathbb K^d$ have a common non-trivial invariant subspace then all operators in the family have a common non-tri…
Helly-Type Theorems for Splitting Point Sets
该研究针对ℝ^d中有限点集的分裂族建立Helly型定理，突破经典准则仅适用于点和超平面的局限，可扩展至任意维数0≤k≤d-1的k维平面族分裂点集的情形。
A Polynomial Kernel for Proper Helly Circular-arc Vertex Deletion
针对 Proper Helly Circular-arc Vertex Deletion 问题，本文在已有 FPT 算法的基础上，提出了一个多项式核。
New Helly-type results for discrete boxes: Quantitative colorful and $(p,q)$-variants
本文研究了$2d-1$族的定量彩色Halman问题及其$(p,q)$型变体，证明了在横截交集包含至少$n$个点的条件下，存在特定盒族使每个盒包含大小受限子集的至少$n$个点。
We consider sets and maps defined over an o-minimal structure over the reals, such as real semi-algebraic or subanalytic sets. A {\em monotone map} is a multi-dimensional generalization of a usual univariate monotone function, while the closure of the graph of a monotone map is a generalization of a compact convex set. In a particular case of an identically constant function, such a graph is called a {\em semi-monotone set}. Graphs of monotone maps are, generally, non-convex, and their intersections, unlike intersections of convex sets, can be topologically complicated. In particular, such an …
Fractional Helly property and combinatorics of forking in NTP$_2$ theories
本文研究了满足分数Helly性质的理论，这些理论扩展了NIP理论，并形成了一种新的低NTP₂理论子类。文章给出了许多新例子，并探讨了分叉和f-通用性在可计算群中的性质，同时提出了关于NTP₂理论中分叉有限组合性质的猜想。
We prove a no-dimensional version of Carathédory's theorem: given an $n$-element set $P\subset \Re^d$, a point $a \in \conv P$, and an integer $r\le d$, $r \le n$, there is a subset $Q\subset P$ of $r$ elements such that the distance between $a$ and $\conv Q$ is less than $\diam P/\sqrt {2r}$. A general no-dimension Helly type result is also proved with colourful and fractional consequences. Similar versions of Tverberg's theorem and some of their extensions are also established.
This paper transfers a randomized algorithm, originally used in geometric optimization, to computational problems in commutative algebra. We show that Clarkson's sampling algorithm can be applied to two problems in computational algebra: solving large-scale polynomial systems and finding small generating sets of graded ideals. The cornerstone of our work is showing that the theory of violator spaces of Gärtner et al.\ applies to polynomial ideal problems. To show this, one utilizes a Helly-type result for algebraic varieties. The resulting algorithms have expected runtime linear in the number …
The partial representation extension problem generalizes the recognition problem for classes of graphs defined in terms of vertex representations. We exhibit circular-arc graphs as the first example of a graph class where the recognition is polynomially solvable while the representation extension problem is NP-complete. In this setting, several arcs are predrawn and we ask whether this partial representation can be completed. We complement this hardness argument with tractability results of the representation extension problem on various subclasses of circular-arc graphs, most notably on all v…
Characterizing the graph classes such that, on $n$-vertex $m$-edge graphs in the class, we can compute the diameter faster than in ${\cal O}(nm)$ time is an important research problem both in theory and in practice. We here make a new step in this direction, for some metrically defined graph classes. Specifically, a subgraph $H$ of a graph $G$ is called a retract of $G$ if it is the image of some idempotent endomorphism of $G$. Two necessary conditions for $H$ being a retract of $G$ is to have $H$ is an isometric and isochromatic subgraph of $G$. We say that $H$ is an absolute retract of some …
Computing the Helly Number, Radon Number and Rank in Cycle Convexity
本文研究循环凸性下图的 Helly 数、Radon 数和秩的计算复杂性与极值结构，证明相关阈值决策问题在平面图中仍为 NP-难和 W[1]-难，并刻画参数取 n-1 和 n-2 的图类。
By extending the definition of boxicity, we extend a Helly-type result given by Danzer and Grumbaum on 2-piercings of family of boxes in $d$-dimensional Euclidian space by lowering the dimension of the boxes in the ambient space.
Multi-graded generic initial ideals, regularity, and the optimal colorful fractional Helly theorem for $d$-Leray complexes
该研究针对多级通用初始理想的正则性问题，基于特定单项式序证明了R/I正则性的下界，并将其应用得出d-Leray单纯复形的最优彩色分数Helly定理。
The classical Kirszbraun theorem says that all $1$-Lipschitz functions $f:A\longrightarrow \mathbb{R}^n$, $A\subset \mathbb{R}^n$, with the Euclidean metric have a $1$-Lipschitz extension to $\mathbb{R}^n$. For metric spaces $X,Y$ we say that $Y$ is $X$-Kirszbraun if all $1$-Lipschitz functions $f:A\longrightarrow Y$, $A\subset X$, have a $1$-Lipschitz extension to~$X$. We analyze the case when $X$ and $Y$ are graphs with the usual path metric. We prove that $\mathbb{Z}^d$-Kirszbraun graphs are exactly graphs that satisfies a certain Helly property. We also consider complexity aspects of these…
We prove several exact quantitative versions of Helly's and Tverberg's theorems, which guarantee that a finite family of convex sets in $R^d$ has a large intersection. Our results characterize conditions that are sufficient for the intersection of a family of convex sets to contain a "witness set" which is large under some concave or log-concave measure. The possible witness sets include ellipsoids, zonotopes, and $H$-convex sets. Our results also bound the complexity of finding the best approximation of a family of convex sets by a single zonotope or by a single $H$-convex set. We obtain colo…
Bounding Radon numbers via Betti numbers
本文利用同调复杂度与 Betti 数界定欧氏空间和曲面上集合族的 Radon 数，并推出分数 Helly、弱 ε-网及 $(p,q)$ 定理。
On colorful generalizations of the Goodman--Pollack transversal problem
本文通过颜色和更一般地，通过组合学的方法，解决了Goodman和Pollack关于凸集族中存在一个F-仿射k维截面的问题，统一了多个经典和最新定理，并提出了颜色和组合学的Dol'nikov型截面定理。
A survey of edge-spectral-Turán type problems in spectral graph theory: Results, conjectures and open problems
本文综述了谱图理论中边缘-谱-图论类型问题的研究成果，提出了相关猜想与开放问题。
We use a geometric description of $m$-cluster categories of Dynkin type $A$ to count the the number of coloured quivers in the $m$-mutation class of quivers of Dynkin type $A$. This is related to angulations of polygons and the cell-growth problem.
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