LEAPS MPS: The Erdos-Ko-Rado Property of Well-Covered Graphs

许多类型的关系和过程，包括物理和社会系统，都可以使用图来建模。这类系统的图模型往往非常大，需要能够在较小的局部级别上从图中提取全局信息的数学技术。极值图论可以被认为是研究图的全局性质如何影响其局部结构。这个项目的目的是研究极值图论中的问题，特别是那些与著名的极值集理论结果Erdos-Ko-Rado定理有关的问题。PI的西班牙裔服务机构的本科生研究人员将结对工作，承担该项目的部分工作。这些学生将有机会学习如何在进行前沿研究的同时利用他们的个人优势。此外，将在该项目中建立一系列讨论会，将PI部门的学生和教师与数学科学领域的高影响力榜样联系起来。Erdos-Ko-Rado （EKR）定理是极值集理论中的一个重要结果，它给出了大小固定且两两相交的集合数目的上界。特别有趣的是一个相交族的直接构造，它通过收集包含一些固定元素的指定大小的所有集合来达到这个界限。在2005年，Holroyd, Spencer， and Talbot为与独立集合的相交族相关的图制定了一个EKR性质。这个性质有一个相应的构造，叫做r星，它取所有大小为r的独立集合，包含图的一个固定顶点。如果大小为r的独立集合的相交族的最大大小等于图中最大的r星的大小，则图称为r- ekr。本课题旨在研究某些图类的r-EKR性质及其相关概念。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。

Many types of relations and processes, including physical and social systems, can be modeled using a graph. Graph models of such systems tend to be very large, requiring mathematical techniques that can extract global information from the graph at the smaller, local level. Extremal graph theory can be thought of as the study of how global properties of a graph influence its local structure. The aim of this project is to investigate questions in extremal graph theory, particularly those that relate to a well-known extremal set theory result called the Erdos-Ko-Rado theorem. Undergraduate student researchers at the PI’s Hispanic-serving institution will work in pairs to take on parts of this project. These students will have the opportunity to learn how to leverage their individual strengths while conducting cutting-edge research. Additionally, a colloquium series will be established within this project to connect students and faculty in the PI’s department to high-impact role models in the mathematical sciences.The Erdos-Ko-Rado (EKR) theorem is a pivotal result in extremal set theory that gives an upper bound on the number of sets of a fixed size that are pairwise intersecting. Of particular interest is the straightforward construction of an intersecting family that attains this bound by collecting all sets of the specified size that contain some fixed element. In 2005, Holroyd, Spencer, and Talbot formulated an EKR property for graphs related to intersecting families of independent sets. This property has a corresponding construction, called an r-star, that takes all independent sets of size r containing a fixed vertex of the graph. A graph is called r-EKR if the maximum size of an intersecting family of size r independent sets is equal to the size of the largest r-star in the graph. This project aims to study the r-EKR property, and related concepts, for certain classes of graphs.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.