Extremal Combinatorics in Geometry and Graph Theory




Beagley, Jonathan Edward

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We study a problem in extremal geometry posed by Paul Erdos and George Szekeres in 1935. This problem is to find the smallest positive integer <italic>N(n)</italic> such that every point set in general position (no three on a line) of <italic>N(n)</italic> points contains the vertex set of a convex <italic>n</italic>-gon. Erdos and Szekeres showed that <italic>N(n)</italic> exists and conjectured that <italic>N(n)</italic> = 2<super><italic>n</italic>-2</super> + 1. In 2006, Walter Morris introduced a graph on the copoints of a planar point set in general position, where cliques in the graph correspond to subsets of points in convex position, and showed that the chromatic number of the copoint graph was <italic>n</italic> if the point set contained at least 2<super><italic>n</italic>-2</super> + 1 points. We extend this copoint graph to abstract convex geometries studied by Edelman and Jamison, where the cliques of this graph are convexly independent sets. A major goal of this dissertation is to study the clique and chromatic numbers for the copoint graph of convex geometries. Much of this dissertation would be trivial if every graph were a copoint graph, so we provide a family of graphs that are not copoint graphs for any convex geometries.



Mathematics, Convex geometry, Copoint graph, Erdos-Szekeres Problem, Graph coloring, Happy Ending Problem, Order dimension