Jérémy Blanc
It is well-known that affine (respectively projective) simple arrangements of n pseudo-lines may have at most n(n-2)/3 (respectively n(n-1)/3) triangles. However, these bounds are reached for only some values of n (mod 6). We provide the best polynomial bound for the affine and the projective case, and for each value of n (mod 6).
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A strong triangle blocking arrangement is a geometric arrangement of some line segments in a triangle with certain intersection properties. It turns out that they are closely related to blocking sets. Our aim in this paper is to prove a classification theorem for strong triangle blocking arrangements. As an application, we obtain a new proof of the result of Ackerman, Buchin, Knauer, Pinchasi and Rote which says that $n$ points in general position cannot be blocked by $n-1$ points, unless $n = 2,4$. We also conjecture an extremal variant of the blocking points problem.
We give some new advances in the research of the maximum number of triangles that we may obtain in a simple arrangements of n lines or pseudo-lines.
Flips of diagonals in colored triangle-free triangulations of a convex polygon are interpreted as moves between two adjacent chambers in a certain graphic hyperplane arrangement. Properties of geodesics in the associated flip graph are deduced. In particular, it is shown that: (1) every diagonal is flipped exactly once in a geodesic between distinguished pairs of antipodes; (2) the number of geodesics between these antipodes is equal to twice the number of Young tableaux of a truncated shifted staircase shape.
Constructing Optimal Kobon Triangle Arrangements via Table Encoding, SAT Solving, and Heuristic Straightening
本文提出通过表格编码、启发式拉直和SAT求解构建最优Kobon三角形排列的新方法，成功发现23线和27线等新最优解，并证明11线情况无解。
A widely investigated subject in combinatorial geometry, originated from Erdős, is the following. Given a point set $P$ of cardinality $n$ in the plane, how can we describe the distribution of the determined distances? This has been generalized in many directions. In this paper we propose the following variants. Consider planar arrangements of $n$ lines. Determine the maximum number of triangles of unit area, maximum area or minimum area, determined by these lines. Determine the minimum size of a subset of these $n$ lines so that all triples determine distinct area triangles. We prove that the…
The $18\cdot 2^t+1$ Triangle-Maximal Series of Straight Lines
研究$18\cdot 2^t+1$条直线的排列中最大有界三角形数量，通过构造19条直线的affine排列，得到无限序列的极大三角形数。
On Triangles in Colored Pseudoline Arrangements
研究双色伪直线排列中是否存在双色三角形，并证明了相关超图的独立数上限。
It is known that cyclic arrangements are the only {\em unavoidable} simple arrangements of pseudolines: for each fixed $m\ge 1$, every sufficiently large simple arrangement of pseudolines has a cyclic subarrangement of size $m$. In the same spirit, we show that there are three unavoidable arrangements of pseudocircles.
Fine Mixed Subdivisions of a Dilated Triangle
本文研究了扩张三角形的细分问题，提出判断唯一划分的标准，并探讨了不同排列间的划分连接操作。
A pseudoline arrangement graph is a planar graph induced by an embedding of a (simple) pseudoline arrangement. We study the corresponding graph realization problem and properties of pseudoline arrangement graphs. In the first part, we give a simple criterion based on the degree sequence that says whether a degree sequence will have a pseudoline arrangement graph as one of its realizations. In the second part, we study the eccentricities of vertices in such graphs. We observe that the diameter (maximum eccentricity of a vertex in the graph) of any pseudoline arrangement graph on $n$ pseudolines…
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.…
Arrangements of Pseudocircles: On Digons and Triangles
本文研究两两相交伪圆简单排列的胞腔结构，证明了 Grünbaum 关于二角形或触点最大数量猜想在含三两两相触伪圆时成立，并构造无此结构的极值非柱面排列；同时证明无二角形和触点排列中三角形最小数为 $\lceil \frac{4}{3}n \rceil$。
We compute all isomorphism classes of simplicial arrangements in the real projective plane with up to 27 lines. It turns out that Grünbaums catalogue is complete up to 27 lines except for four new arrangements with 22, 23, 24, 25 lines, respectively. As a byproduct we classify simplicial arrangements of pseudolines with up to 27 lines. In particular, we disprove Grünbaums conjecture about unstretchable arrangements with at most 16 lines, and prove the conjecture that any simplicial arrangement with at most 14 pseudolines is stretchable.
Let L be a simple Euclidean arrangement of n pseudolines. It is shown that if L has exactly one (>=5)=gon P, and k is the number of edges of P that are adjacent to an unbounded cell of the subarrangement of L induced by the pseudolines in P, then L has exactly n-k triangles and k+n(n-5)/2 quadrilaterals. We also prove that if each pseudoline of L is adjacent to P then L is stretchable.
Enumeration and Classification of Triangle-Maximal Pseudoline Arrangements
该研究提出算法枚举分类最大化三角面的奇数n条伪线简单排列，通过深度优先搜索结合几何剪枝实现，给出不同n的完整枚举结果及大n的部分结果，证明搜索分类的完整性。
In arrangements of pseudocircles (Jordan curves) the weight of a vertex (intersection point) is the number of pseudocircles that contain the vertex in its interior. We give improved upper bounds on the number of vertices of weight <=k in certain arrangements of pseudocircles in the plane. In particular, forbidding certain subarrangements we improve the known bound of 6n-12 (cf. Kedem et al., 1986) for vertices of weight 0 in arrangements of n pseudocircles to 4n-6. In complete arrangements (i.e. arrangements with each two pseudocircles intersecting) we identify two subarrangements of three and…
We study the combinatorics of pseudoline arrangements in the real projective plane. Our focus lies on two classes of arrangements: simplicial arrangements and arrangements whose characteristic polynomials have only real roots. We derive inequalities involving the $t$-vectors of the arrangements in consideration. As application, we obtain some finiteness and classification results. Moreover, we are able to prove the Dirac Motzkin Conjecture for real pseudoline arrangements whose charateristic polynomials split over $\mathbb{R}$.
We describe an incremental algorithm to enumerate the isomorphism classes of double pseudoline arrangements. The correction of our algorithm is based on the connectedness under mutations of the spaces of one-extensions of double pseudoline arrangements, proved in this paper. Counting results derived from an implementation of our algorithm are also reported.
Felsner, Hurtado, Noy and Streinu (2000) conjectured that arrangement graphs of simple great-circle arrangements have chromatic number at most $3$. Motivated by this conjecture, we study the colorability of arrangement graphs for different classes of arrangements of (pseudo-)circles. In this paper the conjecture is verified for $\triangle$-saturated pseudocircle arrangements, i.e., for arrangements where one color class of the 2-coloring of faces consists of triangles only, as well as for further classes of (pseudo-)circle arrangements. These results are complemented by a construction which ma…
$k$-arrangements of pseudolines and pseudocircles
本文证明对每个$m\ge 3$，大小为$m$的不可避免的简单$k$-伪直线排列数量随$k$指数增长，且对偶数$k$，伪圆的$k$-排列也有类似结果。
We study a non-trivial extreme case of the orchard problem for $12$ pseudolines and we provide a complete classification of pseudoline arrangements having $19$ triple points and $9$ double points. We have also classified those that can be realized with straight lines. They include new examples different from the known example of Böröczky. Since Melchior's inequality also holds for arrangements of pseudolines, we are able to deduce that some combinatorial point-line configurations cannot be realized using pseudolines. In particular, this gives a negative answer to one of Grünbaum's problems. We…
Ordered matchings versus triangles via pseudorandom triangle-free graphs
该研究针对有序图的反 Ramsey 数，证明区间色数为 2 的均匀随机有序匹配与三角形形成的反 Ramsey 数下界为 Ω(n^{4/3}/(log n)^{1/3})，通过伪随机无三角图、粗编码和 Brègman 不等式改进现有结果。
In this note we study the logarithmic derivation module of a non-free arrangement. We prove a generalized addition theorem for all arrangements. This addition theorem allows us to find various relationships between non-free arrangements, free arrangements and restriction counts. For graphic arrangements we can use these results to find a lower bound for the maximal degree generator in terms of triangles in the associated graph. We also apply these results to the case of hypersolvable arrangements where we define hyperexponents and use them to find a lower bound for their maximal degree generat…
Flip Graph Connectivity for Arrangements of Pseudolines and Pseudocircles
本文研究伪线和伪圆排列的翻转图连通性，证明伪线翻转图连通性等于最小度n-2，伪圆翻转图在相交排列及柱面相交排列上连通，并证明两类翻转图直径均为Θ(n^3)。
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。
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。
Arrangements of lines and pseudolines are fundamental objects in discrete and computational geometry. They also appear in other areas of computer science, such as the study of sorting networks. Let $B_n$ be the number of nonisomorphic arrangements of $n$ pseudolines and let $b_n=\log_2{B_n}$. The problem of estimating $B_n$ was posed by Knuth in 1992. Knuth conjectured that $b_n \leq {n \choose 2} + o(n^2)$ and also derived the first upper and lower bounds: $b_n \leq 0.7924 (n^2 +n)$ and $b_n \geq n^2/6 -O(n)$. The upper bound underwent several improvements, $b_n \leq 0.6988\, n^2$ (Felsner, 1…
Extremal structure in dense arrangements of $k$-intersecting curves
该研究探讨了在排除完全局部 incidences 模式的情况下，稠密k-相交曲线集的交数上限，并证明了在特定条件下可以改进Pach和Sharir的上界。
Let $E(k, \ell)$ denote the smallest integer such that any set of at least $E(k, \ell)$ points in the plane, no three on a line, contains either an empty convex polygon with $k$ vertices or an empty pseudo-triangle with $\ell$ vertices. The existence of $E(k, \ell)$ for positive integers $k, \ell\geq 3$, is the consequence of a result proved by Valtr [Discrete and Computational Geometry, Vol. 37, 565--576, 2007]. In this paper, following a series of new results about the existence of empty pseudo-triangles in point sets with triangular convex hulls, we determine the exact values of $E(k, 5)$ a…
An arrangement of circles in which circles intersect only in angles of $π/2$ is called an \emph{arrangement of orthogonal circles}. We show that in the case that no two circles are nested, the intersection graph of such an arrangement is planar. The same result holds for arrangement of circles that intersect in an angle of at most $π/2$. For the general case we prove that the maximal number of edges in an intersection graph of an arrangement of orthogonal circles lies in between $4n - O\left(\sqrt{n}\right)$ and $\left(4+\frac{5}{11}\right)n$, for $n$ being the number of circles. Based on the …
We investigate the line arrangement that results from intersecting d complete flags in C^n. We give a combinatorial description of the matroid T_{n,d} that keeps track of the linear dependence relations among these lines. We prove that the bases of the matroid T_{n,3} characterize the triangles with holes which can be tiled with unit rhombi. More generally, we provide evidence for a conjectural connection between the matroid T_{n,d}, the triangulations of the product of simplices Delta_{n-1} x Δ_{d-1}, and the arrangements of d tropical hyperplanes in tropical (n-1)-space. Our work provides a …
Improved Lower Bound on the Number of Pseudoline Arrangements
本文通过将 Dumitrescu 和 Mandal 的伪线排列构造拆分为常数大小片段并用软件计算贡献，将伪线排列数量的下界常数从 0.2083 改进到 0.2604。
With sufficient time, double edge-swap Markov chain Monte Carlo (MCMC) methods are able to sample uniformly at random from many different and important graph spaces. For instance, for a fixed degree sequence, MCMC methods can sample any graph from: simple graphs; multigraphs (which may have multiedges); and pseudographs (which may have multiedges and/or multiple self-loops). In this note we extend these MCMC methods to `multiloop-graphs', which allow multiple self-loops but not multiedges and `loopy-multigraphs' which allow multiedges and single self-loops. We demonstrate that there are degree…
We show that a set of $n$ algebraic plane curves of constant maximum degree can be cut into $O(n^{3/2}\operatorname{polylog} n)$ Jordan arcs, so that each pair of arcs intersect at most once, i.e., they form a collection of pseudo-segments. This extends a similar (and slightly better) bound for pseudo-circles due to Marcus and Tardos. Our result is based on a technique of Ellenberg, Solymosi and Zahl that transforms arrangements of plane curves into arrangements of space curves, so that lenses (pairs of subarcs of the curves that intersect at least twice) become vertical depth cycles. We then …
A classic problem in matroid theory is to find subspace arrangements, specifically hyperplane and pseudosphere arrangements, whose intersection posets are isomorphic to a prescribed geometric lattice. Engström recently showed how to construct an infinite family of such subspace arrangements, indexed by the set of finite regular CW complexes. In this note, we compute the face numbers of these representations (in terms of the face numbers of the indexing complexes) and give upper bounds on the total number of faces in these objects. In particular, we show that, for a fixed rank, the total number…
A straight line triangle representation (SLTR) of a planar graph is a straight line drawing such that all the faces including the outer face have triangular shape. Such a drawing can be viewed as a tiling of a triangle using triangles with the input graph as skeletal structure. In this paper we present a characterization of graphs that have an SLTR. The characterization is based on flat angle assignments, i.e., selections of angles of the graph that have size~$π$ in the representation. We also provide a second characterization in terms of contact systems of pseudosegments. With the aid of disc…
A pseudo-triangle is a simple polygon with three convex vertices, and a pseudo-triangulation is a face-to-face tiling of a planar region into pseudo-triangles. Pseudo-triangulations appear as data structures in computational geometry, as planar bar-and-joint frameworks in rigidity theory and as projections of locally convex surfaces. This survey of current literature includes combinatorial properties and counting of special classes, rigidity theoretical results, representations as polytopes, straight-line drawings from abstract versions called combinatorial pseudo-triangulations, algorithms an…
An Improved Lower Bound on the Number of Pseudoline Arrangements
本文利用动态规划和Lindström-Gessel-Viennot引理，将足够大n下伪线排列数量渐近下界常数$b_n$从0.2083改进至0.2721。
For an arrangement of $n$ pseudolines in the real projective plane let us denote by $t_i$ the number of vertices incident to $i$ lines. We obtain a linear on $t_i$ inequality similar to the Hirzebruch one, but with an elementary proof. We present an algorithm for producing lower bounds of the number of regions basing on linear on $t_i$ inequalities like the above-mentioned. Lower bounds arise in connection with Martinov theorem on the set of all possible numbers of regions and we show how the new bounds may be applied in it.
Given a line arrangement $\cal A$ with $n$ lines, we show that there exists a path of length $n^2/3 - O(n)$ in the dual graph of $\cal A$ formed by its faces. This bound is tight up to lower order terms. For the bicolored version, we describe an example of a line arrangement with $3k$ blue and $2k$ red lines with no alternating path longer than $14k$. Further, we show that any line arrangement with $n$ lines has a coloring such that it has an alternating path of length $Ω(n^2/ \log n)$. Our results also hold for pseudoline arrangements.
Tight bounds on the Carathéodory and exchange numbers in $\triangle$-convexity
本文针对三角形凸性空间，指出并修复了Anand等人关于Carathéodory数与交换数上界证明中的漏洞，给出了完整证明、极值图刻画，并精确计算了块图和k-树的交换数。
Parity patterns meet Genocchi numbers, I: four labelings and three bijections
本文通过四种奇偶限制排列标号超平面排列区域，建立双射并推导交织两类Genocchi数的类Seidel三角形，统一了Hetyei与Lazar-Wachs的计数结果。
A venerable problem in combinatorics and geometry asks whether a given incidence relation may be realized by a configuration of points and lines. The classic version of this would ask for algebraic lines over some field or possibly real pseudolines: embedded circles (isotopic to $RP^1$) in the real projective plane. In this paper we investigate whether a configuration is realized by a collection of $2$-spheres embedded, in the symplectic, smooth, or topological (locally flat) categories, in the complex projective plane. We find obstructions to realizability in the topological category, which a…
From among $ {n \choose 3}$ triangles with vertices chosen from $n$ points in the unit square, let $T$ be the one with the smallest area, and let $A$ be the area of $T$. Heilbronn's triangle problem asks for the maximum value assumed by $A$ over all choices of $n$ points. We consider the average-case: If the $n$ points are chosen independently and at random (with a uniform distribution), then there exist positive constants $c$ and $C$ such that $c/n^3 < μ_n < C/n^3$ for all large enough values of $n$, where $μ_n$ is the expectation of $A$. Moreover, $c/n^3 < A < C/n^3$, with probability close …
We consider a two-parameter family of triangles whose $(n,k)$-th entry (counting the initial entry as the $(0,0)$-th entry) is the number of tilings of $N$-boards (which are linear arrays of $N$ unit square cells for any nonnegative integer $N$) with unit squares and $(1,m-1;t)$-combs for some fixed $m=1,2,\dots$ and $t=2,3,\dots$ that use $n$ tiles in total of which $k$ are combs. A $(1,m-1;t)$-comb is a tile composed of $t$ unit square sub-tiles (referred to as teeth) placed so that each tooth is separated from the next by a gap of width $m-1$. We show that the entries in the triangle are co…
Extending simple monotone drawings
该研究扩展了简单单调绘制，证明了Levi扩展引理的变种，并指出扩展圆柱形单调伪段排列的决策问题为NP难。
If $M$ is the complement of a hyperplane arrangement, and $A=H^*(M,\k)$ is the cohomology ring of $M$ over a field of characteristic 0, then the ranks, $ϕ_k$, of the lower central series quotients of $π_1(M)$ can be computed from the Betti numbers, $b_{ii}=\dim_{\k} \Tor^A_i(\k,\k)_i$, of the linear strand in a (minimal) free resolution of $\k$ over $A$. We use the Cartan-Eilenberg change of rings spectral sequence to relate these numbers to the graded Betti numbers, $b'_{ij}=\dim_{\k} \Tor^E_i(A,\k)_j$, of a (minimal) resolution of $A$ over the exterior algebra $E$. From this analysis, we rec…
Regular triangle unions with maximal number of sides
研究n个三角形的并集的最大边数，提出序列A375986并证明其线性上下界，引入三角形转换方法。
The areas of Ramsey theory and random graphs have been closely linked ever since Erdős' famous proof in 1947 that the 'diagonal' Ramsey numbers $R(k)$ grow exponentially in $k$. In the early 1990s, the triangle-free process was introduced as a model which might potentially provide good lower bounds for the 'off-diagonal' Ramsey numbers $R(3,k)$. In this model, edges of $K_n$ are introduced one-by-one at random and added to the graph if they do not create a triangle; the resulting final (random) graph is denoted $G_{n,\triangle}$. In 2009, Bohman succeeded in following this process for a positi…
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