Research Initiation Award: Turan-type problems on partially ordered sets

HBCU-UP的研究启动奖为STEM教师开展研究活动提供支持，以进一步提高他们的研究能力和效率，并帮助提高hbcu的研究和教学水平。该奖项授予克拉克亚特兰大大学（CAU），将建立一个组合学研究项目，旨在推进部分有序集合和不包含它们的结构的研究。该项目将进一步丰富中国农业大学的教学和学习，通过研究培训和机会让代表性不足的少数民族本科生参与体验式学习，提高他们的技能，同时为他们获得高级学位或进入STEM领域的劳动力做好准备。本项目的目标是对Turán-type问题进行研究，扩展部分有序集（偏序集）Turán-type理论的知识。偏序集是自反的、反对称的和可传递的集合。Turán-type偏序集问题研究不包含传票集的𝑛-dimensional布尔格偏序集族的最大大小，称为禁止传票集问题。具体而言，本项目旨在解决冠偏序集等偏序集上的禁传问题，并提出诱导偏序集上的禁传问题和偏序集饱和问题。这个项目将使用Python编程代码来分析布尔晶格中不包含各种偏置集的族的结构属性。然后，分析将用于改进当前已知的技术，如计数和释放参数，以解决各种（诱导）传票的禁止传票集和偏集饱和问题。本项目的研究结果将极大地促进对偏序集和诱导偏序集Turán-type理论的认识和研究。该项目与卡内基梅隆大学合作，将整合组合学基础、Python编程的学习，以及克拉克亚特兰大大学少数族裔本科生的组合学研究。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。

HBCU-UP’s Research Initiation Awards provide support for STEM faculty to pursue research activities to further their research capabilities and effectiveness and help enhance research and teaching at HBCUs. This award to Clark Atlanta University (CAU) will establish a research program in combinatorics aimed at advancing the study of partially ordered sets and structures that do not contain them. This project will further enrich teaching and learning at CAU by engaging underrepresented minority undergraduate students in experiential learning through research training and opportunities to enhance their skill sets while preparing them for advanced degrees or to enter the workforce in STEM fields.The goal of this project is to conduct research on Turán-type problems and expand the knowledge of Turán-type theory on partially ordered sets (posets). Posets are sets which are reflexive, antisymmetric, and transitive. Turán-type poset problems investigate the maximum size of a family in the 𝑛-dimensional Boolean lattice poset that does not contain some subposet, which is called the forbidden subposet problem. Specifically, this project aims to resolve the forbidden subposet problem on posets like the crown poset, and advance the forbidden subposet problem and the poset saturation problem for induced subposets. This project will utilize programming code in Python to analyze the structural properties of families in the Boolean lattice that do not contain various posets. The analyses will then be used to improve currently known techniques like counting and discharging arguments to solve the forbidden subposet and poset saturation problems for various (induced) subposets. The findings of this project will significantly advance the knowledge and study of Turán-type theory for posets and induced posets. In collaboration with Carnegie Mellon University, the proposed project will integrate the learning of combinatorics foundations, programming in Python, and research in combinatorics for underrepresented minority undergraduate students at Clark Atlanta University.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.