Online Algorithms and Extremal Structures

We study several problems in probabilistic and extremal combinatorics. Probabilistic combinatorics is the area of mathematics studying the behaviour of \most" discrete structures in a given family, as opposed to extremal combinatorics, where one is concerned with the behaviour of \the best" structure with respect to a discrete parameter.
The largest branch of probabilistic combinatorics is the field of random graphs. Given a probability distribution on a collection of graphs, we wish to determine whether a graphtheoretic property is likely to arise or not.