On generalizations of Ramsey and size Ramsey numbers

• Grant number: 24/02872-0
• Support Opportunities: Research Grants - Visiting Researcher Grant - International
• Start date: June 01, 2024
• End date: June 15, 2024
• Field of knowledge: Physical Sciences and Mathematics - Computer Science - Theory of Computation
• Principal Investigator: Guilherme Oliveira Mota
• Grantee: Guilherme Oliveira Mota
• Visiting researcher: Joseph Hyde
• Visiting researcher institution: University of Victoria (UVic), Canada
• Host Institution: Instituto de Matemática e Estatística (IME). Universidade de São Paulo (USP). São Paulo , SP, Brazil

Abstract

This is assistance requested for the visit of Joseph Hyde, University of Victoria, Canada, to the Institute of Mathematics and Statistics of USP (IME/USP), from 6/1/2024 to 6/16/2024 (15 days).This project falls into the area of graph theory, focusing on Ramsey theory. Graphs, representing collections of objects or individuals with edges denoting relationships, offer a versatile model applied across diverse fields such as sociology, chemistry, engineering, computer science, and physics. Ramsey theory, named after Frank P. Ramsey, explores the emergence of complete subgraphs within sufficiently large graphs.The Ramsey number, denoted as r(K_t) for t N, signifies the minimum vertices required in a graph to guarantee the emergence of a complete subgraph K_t. Notoriously challenging, the determination of Ramsey numbers has recently seen significant advancements, including improvements on long-standing asymptotic bounds. Researchers often explore variations like the size-Ramsey number, which indicates the minimum edges ensuring the Ramsey property for a complete subgraph.This project aims to investigate a natural generalization of the Ramsey and size-Ramsey numbers, introduced by Sam Spiro at the 2023 Workshop on Ramsey Theory at the University of California San Diego. The research will contribute to the understanding of these graph theoretic properties, offering insights into their applications and implications across various disciplines. (AU)