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Topics in finite graph Ramsey theory

For a positive integer $r$ and graphs $F$, $G$, and $H$, the graph Ramsey arrow notation $F \longrightarrow (G)^H_r$ means that for every $r$-colouring of the subgraphs of $F$ isomorphic to $H$, there exists a subgraph $G'$ of $F$ isomorphic to $G$ such that all the subgraphs of $G'$ isomorphic to $H$ are coloured the same. Graph Ramsey theory is the study of the graph Ramsey arrow and related arrow notations for other kinds of ``graphs" (\emph{e.g.}, ordered graphs, or hypergraphs). This thesis surveys finite graph Ramsey theory, that is, when all structures are finite.

One aspect surveyed here is determining for which $G$, $H$, and $r$, there exists an $F$ such that $F \longrightarrow (G)^H_r$. The existence of such an $F$ is guaranteed when $H$ is complete, whether ``subgraph" means weak or induced, and existence results are also surveyed when $H$ is non-complete. When such an $F$ exists, other aspects are surveyed, such as determining the order of the smallest such $F$, finding such an $F$ in some restricted family of graphs, and describing the set of minimal such $F$'s.

Identiferoai:union.ndltd.org:LACETR/oai:collectionscanada.gc.ca:MWU.1993/2998
Date18 January 2008
CreatorsBorgersen, Robert David
ContributorsGunderson, David (Mathematics), Doob, Michael (Mathematics) Kocay, William (Computer Science) Platt, Craig (Mathematics)
Source SetsLibrary and Archives Canada ETDs Repository / Centre d'archives des thèses électroniques de Bibliothèque et Archives Canada
Languageen_US
Detected LanguageEnglish

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