Infinite combinatorics in graphs

• Grant number: 23/15912-7
• Support Opportunities: Scholarships in Brazil - Master
• Start date: September 01, 2024
• End date: August 31, 2026
• Field of knowledge: Physical Sciences and Mathematics - Mathematics
• Principal Investigator: Leandro Fiorini Aurichi
• Grantee: Luisa Gomes Seixas
• Host Institution: Instituto de Ciências Matemáticas e de Computação (ICMC). Universidade de São Paulo (USP). São Carlos , SP, Brazil

Abstract

Infinite combinatorics is a mathematical field that includes several areas, and has aplications in several others. In this project, we will focus on the study of results on infinite graphs. We will also continue to study the tools used while studying the cycle double cover conjecture and its generalization, the faithful cycle cover conjecture, that started during undergrad, issuing also other concepts that relate to this study. During undergrad, advences where made regarding said conjectures. Those advences can be found on the preprint Limit of cycles and cover conjectures sent as an attachment, developed with professor Leandro Fiorini Aurichi and his doctorate student Paulo Sérgio Farias Magalhães Júnior. During undergrad, the student focused, together with the professor, on several topics regarding infinite combinatorics. One of the topics studied was the application of combinatorial tecniques, such as elementary submodels, to prove propositions on graphs. The main tecnique studied was used to prove that if some results are true for countable graphs, than it is also true por uncountable graphs - like presented in the preprint attetched. The student and the professor have the goal to search and work on situations where the same tecnique can be used. As the student begins her mesters degree on the first semester of 2024, those studies will be continued.