A study of infinite graphs through the problem of describing Unfriendly Partitions

• Grant number: 21/13373-6
• Support Opportunities: Scholarships in Brazil - Master
• Start date: May 01, 2022
• End date: February 29, 2024
• Field of knowledge: Physical Sciences and Mathematics - Mathematics
• Principal Investigator: Leandro Fiorini Aurichi
• Grantee: Lucas Silva Sinzato Real
• Host Institution: Instituto de Ciências Matemáticas e de Computação (ICMC). Universidade de São Paulo (USP). São Carlos , SP, Brazil

Abstract

This research project aims to study properties of infinite graphs via tools developed by Graph and Set Theories to deal with the Unfriendly Partition Conjecture, trivial in the context of graphs with finite set of vertices but rich when formulated in a general scenario.Although presented as a simple question related to graph colouring, the comprehension of the results developed so far about that conjecture lights also other structural properties of infinite graphs. For example, the existence of unfriendly partitions in rayless graphs can be confirmed once established that this graph family is built by a specific hierarchical procedure.About the Unfriendly Partition Conjecture formulated in a general way, on the other hand, many questions can be asked regarding the only counterexamples published so far, presented by the mathematicians Milner and Shelah. These graphs have all its vertices with infinite degree and cardinality greater than or equal to the first limit cardinal greater than the continuum. Consistently, however, it is possible to conclude that there are necessary at least continuum vertices to construct a graph with the properties described by these authors. To study the existence of unfriendly partitions between these two limit scenarios, therefore, the comprehension of tools from fundamentals of mathematics is needed.

Academic Publications

(References retrieved automatically from State of São Paulo Research Institutions)

REAL, Lucas Silva Sinzato. Problem-solving techniques in infinite graphs. 2024. Master's Dissertation - Universidade de São Paulo (USP). Instituto de Ciências Matemáticas e de Computação (ICMC/SB) São Carlos.