Ramsey theory for cycles and paths

Author(s)

Fabricio Siqueira Benevides

• Document type: Master's Dissertation
• Press: São Paulo.
• Institution: Universidade de São Paulo (USP). Instituto de Matemática e Estatística (IME/SBI)
• Defense date: 2007-03-26

Examining board members

Yoshiharu Kohayakawa; Paulo Feofiloff; Carlos Gustavo Tamm de Araujo Moreira

• Advisor: Yoshiharu Kohayakawa

Abstract

The main objects of interest in this work are the Ramsey numbers for cycles and the Szemerédi regularity lemma. For graphs $L_1, \\ldots, L_k$, the Ramsey number $R(L_1, \\ldots,L_k)$ is the minimum integer $N$ such that for any edge-coloring of the complete graph with~$N$ vertices by $k$ colors there exists a color $i$ for which the corresponding color class contains~$L_i$ as a subgraph. We are specially interested in the case where the graphs $L_i$ are cycles. We obtained an original result solving the case where $k=3$ and $L_i$ are even cycles of the same length. (AU)