Ramsey regions and simplicial homology tables for graphs - Frederick, Christopher Austin, author; Peterson, Chris, advisor (Colorado State University. Libraries, 2008)
Ramsey Theory is the investigation of edge-colored graphs which force a monochromatic subgraph. We devise a way of breaking certain Ramsey Theory problems into "smaller" pieces so that information about Ramsey Theory can be gained without solving the entire problem, (which is often difficult to solve). Next the work with Ramsey Regions for graphs is translated into the language of hypergraphs. Theorems and techniques are reworked to fit appropriately into the setting of hypergraphs. The work of persistence complex on large data sets is examined in the setting of graphs. Various simplicial complexes can be assigned to a graph. For a given simplicial complex the persistence complex can be constructed, giving a highly detailed graph invariant. Connections between the graph and persistence complex are investigated.
Keywords: graph theory; homology tables; Ramsey regions; mathematics