Collinear Triple Hypergraphs and the Finite Plane Kakeya Problem

最小化共线三元组数量的置换

Permutations minimizing the number of collinear triples

本文刻画了有限域上使图中共线三元组数量最少的置换，确定字典序最小者，证实 Cooper-Solymosi 猜想，并探讨其与正交拉丁方及有限仿射平面的联系。

Saturating sets in projective planes and hypergraph covers

Let $Π_q$ be an arbitrary finite projective plane of order $q$. A subset $S$ of its points is called saturating if any point outside $S$ is collinear with a pair of points from $S$. Applying probabilistic tools we improve the upper bound on the smallest possible size of the saturating set to $\lceil\sqrt{3q\ln{q}}\rceil+ \lceil(\sqrt{q}+1)/2\rceil$. The same result is presented using an algorithmic approach as well, which points out the connection with the transversal number of uniform multiple intersecting hypergraphs.

Betweenness Structures of Small Linear Co-Size

One way to study the combinatorics of finite metric spaces is to study the betweenness relation associated with the metric space. In the hypergraph metrization problem, one has to find and characterize metric betweennesses whose collinear triples (or alternatively, non-degenerate triangles) coincide with the edges of a given $3$-uniform hypergraph. Metrizability of different kinds of hypergraphs was investigated in the last decades. Chen showed that steiner triple systems are not metrizable, while Richmond and Richmond characterized linear betweennesses, i.e. metric betweennesses that realize …

On the number of points in general position in the plane

In this paper we study some Erdos type problems in discrete geometry. Our main result is that we show that there is a planar point set of n points such that no four are collinear but no matter how we choose a subset of size $n^{5/6+o(1)} $ it contains a collinear triple. Another application studies epsilon-nets in a point-line system in the plane. We prove the existence of some geometric constructions with a new tool, the so-called Hypergraph Container Method.

无等腰网格的稀疏染色：一个AI辅助证明案例研究

Sparse coloring for isosceles-free grids: an AI-assisted proof case study

本研究通过稀疏超图染色定理证明无等腰三元组的网格子集大小下界，改进PatternBoost的保证，并分析AI辅助证明过程中的检查实践。

Exploring the infinitesimal rigidity of planar configurations of points and rods

This article is concerned with the rigidity properties of geometric realizations of incidence geometries of rank two as points and lines in the Euclidean plane; we care about the distance being preserved among collinear points. We discuss the rigidity properties of geometric realizations of incidence geometries in relation to the rigidity of geometric realizations of other well-known structures, such as graphs and hypergraphs.The $2$-plane matroid is also discussed. Further, we extend a result of Whiteley to determine necessary conditions for an incidence geometry of points and lines with exac…