Problems and results in extremal combinatorics - I

Extremal combinatorics is an area in discrete mathematics that has developed spectacularly during the last decades. This paper contains a collection of problems and results in the area, including solutions or partial solutions to open problems suggested by various researchers in extremal graph theory, extremal finite set theory and combinatorial geometry. This is not meant to be a comprehensive survey of the area, it is merely a collection of various extremal problems, which are hopefully interesting. The choice of the problems is inevitably somewhat biased, and as the title of the paper suggests I hope to write a related paper in the future. Each section of this paper is essentially self contained, and can be read separately.