Open Coloring Axiom and coloring applications

Author(s)

Thales Sarinho Galvão Santos de Souza

• Document type: Master's Dissertation
• Press: São Carlos.
• Institution: Universidade de São Paulo (USP). Instituto de Ciências Matemáticas e de Computação (ICMC/SB)
• Defense date: 2023-07-03

Examining board members

Leandro Fiorini Aurichi; Christina Brech; Igor Mencattini; Samuel Gomes da Silva

• Advisor: Leandro Fiorini Aurichi

Abstract

This dissertation explores the Open Coloring Axiom (OCA) and its applications. This axiom was introduced by Todorcevíc and it can be viewed as a two dimensional property of perfect sets. The OCA states that for every open coloring of [S]2 with two colors, there exists an uncountable subset of S that all of its pairs have color 0, or else S can be covered by countably many sets that all of its pais have color 1. Throughout this dissertation, we present applications to the OCA, OCAs relationship with other axioms and we studied ways to generalize its statement. We also studied forcing techniques aiming to prove that OCA is consistent with ZFC. Finally, we present two attachments that gather results involving graphs and the Kuratowski Theorem, and the relationship between CH and Luzins axiom. (AU)

• FAPESP's process: 21/02478-1 - Open Coloring Axiom and other coloring applications
• Grantee: Thales Sarinho Galvão Santos de Souza
• Support Opportunities: Scholarships in Brazil - Master