Chaya Keller, Rom Pinchasi
Let $P$ be a set of $n$ points in general position in the plane. Let $R$ be a set of $n$ points disjoint from $P$ such that for every $x,y \in P$ the line through $x$ and $y$ contains a point in $R$ outside of the segment delimited by $x$ and $y$. We show that $P \cup R$ must be contained in a cubic curve.
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We prove that every finite family of convex sets in the plane satisfying the $(4,3)$-property can be pierced by $9$ points. This improves the bound of $13$ proved by Gyárfás, Kleitman, and Tóth in 2001.
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.
Let $P$ be a set of $n$ points in general position in the plane. Let $R$ be a set of points disjoint from $P$ such that for every $x,y \in P$ the line through $x$ and $y$ contains a point in $R$. We show that if $|R| < \frac{3}{2}n$ and $P \cup R$ is contained in a cubic curve $c$ in the plane, then $P$ has a special property with respect to the natural group action on $c$. That is, $P$ is contained in a coset of a subgroup $H$ of $c$ of cardinality at most $|R|$. We use the same approach to show a similar result in the case where each of $B$ and $G$ is a set of $n$ points in general position …
An $(\aleph_0,k+2)$-Theorem for $k$-Transversals
本文证明了关于$k$-横截的$(\u2135_0,k+2)$-定理：若无限闭球族中每可数多个元素有$k+2$个可被$k$维平面刺穿，则整个族可被有限个$k$维平面刺穿，并推广到近球族。
Bounds on piercing and line-piercing numbers in families of convex sets in the plane
本文研究平面紧凸集族的穿刺问题，利用拓扑 KKM 定理给出 (p+1,2) 性质下直线穿刺数及结合交族性质的点穿刺数界，并证明多彩版本。
Improvement on line transversals of families of connected sets in the plane
本文研究平面紧连通集族的直线横截问题，证明任意三集可被一直线刺穿时存在三条共点直线刺穿全族，并给出从6族改进到5族的彩色版本及平行直线推广。
We define a $C(k)$ to be a family of $k$ sets $F_1,\dots,F_k$ such that $\textrm{conv}(F_i\cup F_{i+1})\cap \textrm{conv}(F_j\cup F_{j+1})=\emptyset$ when $\{i,i+1\}\cap \{j,j+1\}=\emptyset$ (indices are taken modulo $k$). We show that if $\mathcal{F}$ is a family of compact, convex sets that does not contain a $C(k)$, then there are $k-2$ lines that pierce $\mathcal{F}$. Additionally, we give an example of a family of compact, convex sets that contains no $C(k)$ and cannot be pierced by $\left\lceil \frac{k}{2} \right\rceil -1$ lines.
No Infinite $(p,q)$-Theorem for Piercing Compact Convex Sets with Lines in $\mathbb{R}^3$
通过构造反例，证明在三维空间中不存在关于直线刺穿紧凸集的无限 $(p,q)$-定理，即无限子集可被直线刺穿任意有限个集合，但整体无有限直线横截。
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。
Transversals to colorful intersecting convex sets
本文证明Dolnikov猜想在常宽体或接近圆盘情形下的更强版本，并给出8点穿透的一般结果，同时优化了二维情形下穿透点与横截线的数量。
A family of sets $F$ is said to satisfy the $(p,q)$ property if among any $p$ sets in $F$, some $q$ have a non-empty intersection. Hadwiger and Debrunner (1957) conjectured that for any $p \geq q \geq d+1$ there exists $c=c_d(p,q)$, such that any family of compact convex sets in $\mathbb{R}^d$ that satisfies the $(p,q)$ property, can be pierced by at most $c$ points. In a celebrated result from 1992, Alon and Kleitman proved the conjecture. However, obtaining sharp bounds on $c_d(p,q)$, called `the Hadwiger-Debrunner numbers', is still a major open problem in discrete and computational geometr…
For a finite set $A\subset \mathbb{R}^d$, let $Δ(A)$ denote the spread of $A$, which is the ratio of the maximum pairwise distance to the minimum pairwise distance. For a positive integer $n$, let $γ_d(n)$ denote the largest integer such that any set $A$ of $n$ points in general position in $\mathbb{R}^d$, satisfying $Δ(A) \leq αn^{1/d}$ for a fixed $α>0$, contains at least $γ_d(n)$ points in convex position. About $30$ years ago, Valtr proved that $γ_2(n)=Θ(n^{1/3})$. Since then no further results have been obtained in higher dimensions. Here we continue this line of research in three dimensi…
We give improved lower bounds on the minimum number of $k$-holes (empty convex $k$-gons) in a set of $n$ points in general position in the plane, for $k=5,6$.
A point set $M$ in the Euclidean plane is called a planar integral point set if all the distances between the elements of $M$ are integers, and $M$ is not situated on a straight line. A planar integral point set is called to be in semi-general position, if it does not contain collinear triples. The existing lower bound for mininum diameter of planar integral point sets is linear. We prove a new lower bound for mininum diameter of planar integral point sets in semi-general position that is better than linear.
The general position number ${\rm gp}(G)$ of a connected graph $G$ is the cardinality of a largest set $S$ of vertices such that no three distinct vertices from $S$ lie on a common geodesic; such sets are refereed to as gp-sets of $G$. The general position number of cylinders $P_r\,\square\, C_s$ is deduced. It is proved that ${\rm gp}(C_r\,\square\, C_s)\in \{6,7\}$ whenever $r\ge s \ge 3$, $s\ne 4$, and $r\ge 6$. A probabilistic lower bound on the general position number of Cartesian graph powers is achieved. Along the way a formula for the number of gp-sets in $P_r\,\square\, P_s$, where $r…
Edge general position sets in Fibonacci and Lucas cubes
本文研究偏立方体的边一般位置集，证明 Fibonacci 立方体与 Lucas 立方体中两个最大 Θ-类的并集构成极大边一般位置集。
Erdős asked the following question: given $n$ points in the plane in almost general position (no 4 collinear), how large a set can we guarantee to find that is in general position (no 3 collinear)? Füredi constructed a set of $n$ points in almost general position with no more than $o(n)$ points in general position. Cardinal, Tóth and Wood extended this result to $\mathbb{R}^3$, finding sets of $n$ points with no 5 on a plane whose subsets with no 4 points on a plane have size $o(n)$, and asked the question for higher dimensions: for given $n$, is it still true that the largest subset in genera…
Counting largest mutual-visibility and general position sets of glued $t$-ary trees
研究粘合 $t$-ary 树的最大互视集和一般位置集的计数问题，确定相关不变量并扩展至更一般的情况。
General position sets, colinear sets, and Sierpiński product graphs
研究了西尔皮ński积图中一般位置集和共线集的性质，推导了相关数的公式并给出了界限。
The general position problem in graph theory asks for the largest set $S$ of vertices of a graph $G$ such that no shortest path of $G$ contains more than two vertices of $S$. In this paper we consider a variant of the general position problem called the \emph{monophonic position problem}, obtained by replacing `shortest path' by `induced path'. We prove some basic properties and bounds for the monophonic position number of a graph and determine the monophonic position number of some graph families, including unicyclic graphs, complements of bipartite graphs and split graphs. We show that the m…
The general $d$-position number ${\rm gp}_d(G)$ of a graph $G$ is the cardinality of a largest set $S$ for which no three distinct vertices from $S$ lie on a common geodesic of length at most $d$. This new graph parameter generalizes the well studied general position number. We first give some results concerning the monotonic behavior of ${\rm gp}_d(G)$ with respect to the suitable values of $d$. We show that the decision problem concerning finding ${\rm gp}_d(G)$ is NP-complete for any value of $d$. The value of ${\rm gp}_d(G)$ when $G$ is a path or a cycle is computed and a structural charac…
On the Largest Convexity Number of Co-Finite Sets in the Plane
研究平面中非有限集的最大凸性数，证明了在一般位置和凸位置点集下的凸性数界。
Monophonic position sets of Cartesian and lexicographic products of graphs
本文针对图的笛卡尔积与字典积，研究将一般位置问题中最短路径替换为诱导路径的单音位置问题，推导两类积图单音位置数的相关结论与计算公式。
We consider point sets in the real projective plane $\mathbb{R}P^2$ and explore variants of classical extremal problems about planar point sets in this setting, with a main focus on Erdős--Szekeres-type problems. We provide asymptotically tight bounds for a variant of the Erdős--Szekeres theorem about point sets in convex position in $\mathbb{R}P^2$, which was initiated by Harborth and Möller in 1994. The notion of convex position in $\mathbb{R}P^2$ agrees with the definition of convex sets introduced by Steinitz in 1913. For $k \geq 3$, an (\affine) $k$-hole in a finite set $S \subseteq \math…
General position sets in strong products with paths and cycles
该研究确定了路径、环与连通图的强积图的一般位置数相关结果，反驳了一般位置数在强积下的可乘性猜想，还将结果应用于双环强积图并构造了无限族。
Given a set of $n$ points in $R^2$, the Szemerédi-Trotter theorem establishes that the number of lines which can be incident to at least $k > 1$ of these points is $O(n^2/k^3 + n/k)$. J.\ Solymosi conjectured that if one requires the points to be in a grid formation and the lines to be in general position---no two parallel, no three meeting at a point---then one can get a much tighter bound. We prove: for every $ε> 0$ there exists some $δ> 0$ such that for sufficiently large values of $n$, every set of lines in general position, each intersecting an $n \times n$ grid of points in at least $n^{…
Extremal edge general position sets in some graphs
本文研究图的边一般位置数，刻画了边一般位置数为边数减一或等于三的图，并给出块图的尖锐上下界及若干特定块图的精确值。
Variety of general position problems in graphs
本文引入并研究图的多种一般位置问题，定义全、对偶和外部一般位置数，刻画其与单纯顶点、最大距离顶点及凸补集的关系，并确定任意笛卡尔积的相应数值。
A $k$-crossing family in a point set $S$ in general position is a set of $k$ segments spanned by points of $S$ such that all $k$ segments mutually cross. In this short note we present two statements on crossing families which are based on sets of small cardinality: (1) Any set of at least 15 points contains a crossing family of size 4. (2) There are sets of $n$ points which do not contain a crossing family of size larger than $8\lceil \frac{n}{41} \rceil$. Both results improve the previously best known bounds.
Colouring a graph with position sets
本文研究图的广义位置着色问题，定义$\gp$-色数，给出其界并确定多类图的精确值，证明该问题是NP完全的。
A vertex subset $S$ of a graph $G$ is a general position set of $G$ if no vertex of $S$ lies on a geodesic between two other vertices of $S$. The cardinality of a largest general position set of $G$ is the general position number ${\rm gp}(G)$ of $G$. It is proved that $S\subseteq V(G)$ is in general position if and only if the components of $G[S]$ are complete subgraphs, the vertices of which form an in-transitive, distance-constant partition of $S$. If ${\rm diam}(G) = 2$, then ${\rm gp}(G)$ is the maximum of $ω(G)$ and the maximum order of an induced complete multipartite subgraph of the co…
Lower General Position Sets in Graphs
本文将M. Gardner问题推广至图论，引入下一般位置数概念，确定多类图该数值并证明其判定问题为NP完全。
A planar point set is in convex position precisely when it has a convex polygonization, that is, a polygonization with maximum interior angle measure at most π. We can thus talk about the convexity of a set of points in terms of the minimum, taken over all polygonizations, of the maximum interior angle. The main result presented here is a nontrivial combinatorial upper bound of this min-max value in terms of the number of points in the set. Motivated by a particular construction, we also pose a natural conjecture for the best upper bound.
Planar Point Sets Determine Many Pairwise Crossing Segments
该研究改进了平面一般位置点集两两相交线段数量的下界，从25多年前的Ω(√n)提升至n^{1-o(1)}，给出构造性证明且结果可推广到稠密几何图。
General position problems in strong and lexicographic products of graphs
本文研究强和字典序图积中的一般位置问题，确定了外积、对偶积和总积的一般位置数，并获得了若干确切值。
For every pattern $P$, consisting of a finite set of points in the plane, $S_{P}(n,m)$ is defined as the largest number of similar copies of $P$ among sets of $n$ points in the plane without $m$ points on a line. A general construction, based on iterated Minkovski sums, is used to obtain new lower bounds for $S_{P}(n,m)$ when $P$ is an arbitrary pattern. Improved bounds are obtained when $P$ is a triangle or a regular polygon with few sides. It is also shown that $S_{P}(n,m)\geq n^{2-ε}$ whenever $m(n)\to \infty$ as $n \to\infty$. Finite sets with no collinear triples and not containing the 4 …
Mutual-visibility and general position sets in Sierpiński triangle graphs
本文研究 Sierpiński 三角形图中的一般位置集与全、外、对偶等互可见集变体，给出了相关最大集合规模的结果。
Lower General Position in Cartesian Products
本文研究笛卡尔积图的下一般位置问题，通过引入终端集概念刻画小极大一般位置集合的存在性，并给出直径至多三的图的构造性证明及相关猜想和下界。
Given a finite set $A \subseteq \mathbb{R}^d$, points $a_1,a_2,\dotsc,a_{\ell} \in A$ form an $\ell$-hole in $A$ if they are the vertices of a convex polytope which contains no points of $A$ in its interior. We construct arbitrarily large point sets in general position in $\mathbb{R}^d$ having no holes of size $O(4^dd\log d)$ or more. This improves the previously known upper bound of order $d^{d+o(d)}$ due to Valtr. The basic version of our construction uses a certain type of equidistributed point sets, originating from numerical analysis, known as $(t,m,s)$-nets or $(t,s)$-sequences, yielding…
On the connectivity of the disjointness graph of segments of point sets in general position in the plane
研究平面一般位置点集的不相交图连通性，证明其连通度下界并指出该界对所有 n≥3 紧。
Random Turán and counting results for general position sets over finite fields
本文研究有限域随机子集中一般位置集的最大规模与计数，利用超图容器及二维点线关联图的伪随机性，给出二维数量级、高维近紧上界和渐近紧计数界。
A flag of a finite set $S$ is a set $f$ of non-empty proper subsets of $S$ such that $A\subseteq B$ or $B\subseteq A$ for all $A,B\in f$. The set $\{|A|:A\in f\}$ is called the type of $f$. Two flags $f$ and $f'$ are in general position (with respect to $S$) when $A\cap B=\emptyset$ or $A\cup B=S$ for all $A\in f$ and $B\in f'$. We study sets of flags of a fixed type $T$ that are mutually not in general position and are interested in the largest cardinality of these sets. This is a generalization of the classical Erdös-Ko-Rado problem. We will give some basic facts and determine the largest ca…
In this paper we generalise the notion of visibility from a point in an integer lattice to the setting of graph theory. For a vertex $x$ of a connected graph $G$, we say that a set $S \subseteq V(G)$ is an \emph{$x$-position set} if for any $y \in S$ the shortest $x,y$-paths in $G$ contain no point of $S\setminus \{ y\}$. We investigate the largest and smallest orders of maximum $x$-position sets in graphs, determining these numbers for common classes of graphs and giving bounds in terms of the girth, vertex degrees, diameter and radius. Finally we discuss the complexity of finding maximum ver…
In 1935, Erdős and Szekeres proved that every set of $n$ points in general position in the plane contains the vertices of a convex polygon of $\frac{1}{2}\log_2(n)$ vertices. In 1961, they constructed, for every positive integer $t$, a set of $n:=2^{t-2}$ points in general position in the plane, such that every convex polygon with vertices in this set has at most $\log_2(n)+1$ vertices. In this paper we show how to realize their construction in an integer grid of size $O(n^2 \log_2(n)^3)$.
Varieties of mutual-visibility and general position on Sierpiński graphs
本文研究了Sierpiński图$S_p^n$上的八类互见性与一般位置不变量，确定了$S_p^2$的全部值及$S_3^n$的大部分值，并枚举了相应极值集合。
Infinite rational distance sets in affine general position: constructions in every dimension
该研究在任意维数下构造了仿射一般位置的无限有理距离集，奇数维时还可实现无 $d+2$ 点共球，三维时得到无限多非相似本原 $n_3$-簇，为相关几何问题提供了统一构造方法。
In this paper we show that every sufficiently large family of convex bodies in the plane has a large subfamily in convex position provided that the number of common tangents of each pair of bodies is bounded and every subfamily of size five is in convex position. (If each pair of bodies have at most two common tangents it is enough to assume that every triple is in convex position, and likewise, if each pair of bodies have at most four common tangents it is enough to assume that every quadruple is in convex position.) This confirms a conjecture of Pach and Toth, and generalizes a theorem of Bi…
On polynomials associated to Voronoi diagrams of point sets and crossing numbers
本文为一般位置点集定义了 Voronoi、圆和 E≤k 三个多项式，用其表达直线交叉数，并证明 Voronoi 多项式根在单位圆上当且仅当点集凸，且给出根的位置界。
Variants of Wythoff's Games with Different Terminal Sets
本文研究了经典威斯霍夫博弈的变种，通过改变终止集定义，利用斐波那契数列描述P态，无需递归。
Two elements, $x$ and $y$, are separated by a set $S$ if it contains exactly one of $x$ and $y$. We prove that any set of $n$ points in general position in the plane can be separated by $O(n\log\log n/\log n)$ convex sets, and for some point sets $Ω(n/\log n)$ convex sets are necessary.
On higher dimensional point sets in general position
本文利用超图容器方法，研究了高维点集的一般位置问题，并改进了相关参数的上界。
A set $S\subseteq\{0,1\}^E$ of binary vectors, with positions indexed by $E$, is said to be a \textit{powerful code} if, for all $X\subseteq E$, the number of vectors in $S$ that are zero in the positions indexed by $X$ is a power of 2. By treating binary vectors as characteristic vectors of subsets of $E$, we say that a set $S\subseteq2^E$ of subsets of $E$ is a \textit{powerful set} if the set of characteristic vectors of sets in $S$ is a powerful code. Powerful sets (codes) include cocircuit spaces of binary matroids (equivalently, linear codes over $\mathbb{F}_2$), but much more besides. O…
There is a simple formula for the Ehrhart polynomial of a cyclic polytope. The purpose of this paper is to show that the same formula holds for a more general class of polytopes, lattice-face polytopes. We develop a way of decomposing any d-dimensional simplex in general position into d! signed sets, each of which corresponds to a permutation in the symmetric group, and reduce the problem of counting lattice points in a polytope in general position to that of counting lattice points in these special signed sets. Applying this decomposition to a lattice-face simplex, we obtain signed sets with …
We study the following family of problems: Given a set of $n$ points in convex position, what is the maximum number triangles one can create having these points as vertices while avoiding certain sets of forbidden configurations. As forbidden configurations we consider all 8 ways in which a pair of triangles in such a point set can interact. This leads to 256 extremal Turán-type questions. We give nearly tight (within a $\log n$ factor) bounds for 248 of these questions and show that the remaining 8 questions are all asymptotically equivalent to Stein's longstanding tripod packing problem.
Suppose that $nk$ points in general position in the plane are colored red and blue, with at least $n$ points of each color. We show that then there exist $n$ pairwise disjoint convex sets, each of them containing $k$ of the points, and each of them containing points of both colors. We also show that if $P$ is a set of $n(d+1)$ points in general position in $\mathbb{R}^d$ colored by $d$ colors with at least $n$ points of each color, then there exist $n$ pairwise disjoint $d$-dimensional simplices with vertices in $P$, each of them containing a point of every color. These results can be viewed a…
An empty pentagon in a point set P in the plane is a set of five points in P in strictly convex position with no other point of P in their convex hull. We prove that every finite set of at least 328k^2 points in the plane contains an empty pentagon or k collinear points. This is optimal up to a constant factor since the (k-1)x(k-1) grid contains no empty pentagon and no k collinear points. The previous best known bound was doubly exponential.
Two sets $A$ and $B$ of points in the plane are \emph{mutually avoiding} if no line generated by any two points in $A$ intersects the convex hull of $B$, and vice versa. In 1994, Aronov, Erd\H os, Goddard, Kleitman, Klugerman, Pach, and Schulman showed that every set of $n$ points in the plane in general position contains a pair of mutually avoiding sets each of size at least $\sqrt{n/12}$. As a corollary, their result implies that for every set of $n$ points in the plane in general position one can find at least $\sqrt{n/12}$ segments, each joining two of the points, such that these segments …
A subset $R\subseteq V(G)$ of a graph $G$ is a general position set if any triple set $R_0$ of $R$ is non-geodesic in $G$, that is, no vertex of $R_0$ lies on any geodesic between the other two vertices of $R_0$ in $G$. Let $\mathcal{R}$ be the set of general position sets of a graph $G$. The general position number of a graph $G$, denoted by $gp(G)$, is defined as $gp(G)=\max\{|R|:R\in\mathcal{R}\}$. In this paper, we determine the bounds on the gp-numbers for any maximal outerplane graph and characterize the corresponding extremal graphs.
We introduce the theory of div point sets, which aims to provide a framework to study the combinatoric nature of any set of points in general position on an Euclidean plane. We then show that proving the unsatisfiability of some first-order logic formulae concerning some sets of multisets of uniform cardinality over boolean variables would prove the Erdös-Szekeres conjecture, which states that for any set of 2^(n-2)+1 points in general position, there exists n points forming a convex polygon, where n is greater than or equal to 3.
Let $B$ and $R$ be point sets (of {\em blue} and {\em red} points, respectively) in the plane, such that $P:=B\cup R$ is in general position, and $|P|$ is even. A line $\ell$ is {\em balanced} if it spans one blue and one red point, and on each open halfplane of $\ell$, the number of blue points minus the number of red points is the same. We prove that $P$ has at least $\min \{|B|,|R|\} $ balanced lines. This refines a result by Pach and Pinchasi, who proved this for the case $|B|=|R|$.
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