Discrete Geometry and Extremal Combinatorics

这个项目涉及离散几何和极值组合交叉的几个开放问题。传统上，离散几何中的问题涉及点、线、三角形、平面或其他简单几何对象的集合，其中许多问题非常自然，值得研究。其中一些，如三维凸多面体的结构，可以追溯到古代，而另一些也与现代数学的各个不同领域密切相关，特别是极值组合。近年来，这些丰富的互动导致了这两个领域之间的一些显着发展，这个项目的目标是尽可能多地利用这一势头。本项目的第一部分涉及Ramsey理论围绕Erdős-Szekeres问题，该问题是关于在一般位置的点集的有限构型中存在大凸多边形，特别着眼于为图和超图的各种经典Ramsey数建立新的上界。本建议的第二部分是关于关联几何，这是一个植根于Turán-type极值图论问题的领域，它也通过所谓的和积现象与数学的其他分支，如谐波分析和数论，有根本的联系。PI打算通过研究这个故事的不同方面出现的几个新旧自然问题来进一步发展这些联系。激励（长期存在）问题的例子包括：Zarankiewicz问题、单位距离猜想和Heilbronn三角形问题。作为副产品，PI还计划开发新的工具，以进一步促进离散数学中代数、分析、组合和概率方法之间的相互作用。PI计划让研究生参与这个项目。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

This project concerns several open problems at the intersection of discrete geometry and extremal combinatorics. Questions in discrete geometry traditionally involve sets of points, lines, triangles, planes, or other simple geometric objects, and many of them are tantalizingly natural and worth studying for their own sake. Some of them, such as the structure of 3-dimensional convex polytopes, go back to the antiquity, while others are also intimately connected with various different areas of modern mathematics, in particular extremal combinatorics. In recent years, these rich interactions have led to several remarkable developments between these two fields, and the goal of this project is to essentially capitalize as much as possible on this momentum. The first part of this project concerns Ramsey theory around the Erdős-Szekeres problem about the existence of large convex polytopes in finite configurations of point sets in general position, with an eye particularly towards establishing new upper bounds for various classical Ramsey numbers for graphs and hypergraphs. The second part of this proposal is about incidence geometry, an area with roots in Turán-type problems in extremal graph theory which is also fundamentally connected with other branches of mathematics, such as harmonic analysis and number theory, via the so-called sum-product phenomenon. The PI intends to further develop these connections by studying several old and new natural problems that arise on the different sides of this story. Examples of motivating (longstanding) questions include: the Zarankiewicz problem, the unit distance conjecture, and the Heilbronn triangle problem. As a byproduct, the PI also plans to develop new tools that could further the interplay between algebraic, analytic, combinatorial, and probabilistic methods in discrete mathematics. The PI plans to involve graduate students in this project.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.