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09/30/05 Colloquium
Dr. Greg McColm
University of South Florida
University of South Florida
Thresholds in Random Graph Theory
Abstract: The "evolution of a random graph" is any stochastic process by which edges are added "randomly" to a graph; initially consisting of isolated vertices, the graph evolves towards some terminal graph. If an upwards closed property is one which, once true of a graph, remains true if edges are added to that graph, then we can ask, "when will the property become true in the evolution of a random graph?"The trickiness of this question is reflected in the fact that, for some upwards closed properties, the question is not "when do we expect the property to become true?" even though this would seem to be the obvious probabilistic question. Recently some quite simple counterexamples have exploded some hitherto unexamined prejudices, and should force a belated examination of what is going on, and so we look at the issues and complications involved. Although this subject lies in combinatorics and probability, this presentation is aimed at the colloquium level and should be generally accessible.
