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。这种系统的图模型往往非常大，需要数学技术，可以从较小的局部层次上的图中提取全局信息。极值图论可以被认为是研究图的全局属性如何影响其局部结构的理论。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有关的问题，称为Erdos-Ko-Rado定理。PI的西班牙裔服务机构的本科生研究人员将成对工作，承担该项目的一部分。这些学生将有机会学习如何利用自己的个人优势，同时进行前沿研究。此外，还将在该项目中建立一个系列讨论会，将PI部门的学生和教师与数学科学中的高影响力角色模型联系起来。Erdos-Ko-Rado（EKR）定理是极值集理论中的一个关键结果，它给出了固定大小的两两相交的集合数量的上限。特别感兴趣的是一个相交的家庭，通过收集所有集的指定大小，包含一些固定的元素，达到这一界限的直接建设。在2005年，Holroyd，Spencer和塔尔博特提出了一个与独立集相交族相关的图的EKR性质。这个性质有一个对应的构造，称为r-星，它取所有包含图的固定顶点的大小为r的独立集。一个图称为r-EKR，如果一个相交的r大小独立集族的最大大小等于图中最大的r-星的大小。该项目旨在研究r-EKR属性和相关概念，为某些类别的graphs.This奖项反映了NSF的法定使命，并已被认为是值得的支持，通过评估使用基金会的知识价值和更广泛的影响审查标准。

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.