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

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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.