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Extremal combinatorics

Extremal combinatorics studies how large or how small a collection of finite objects (numbers, graphs, vectors, sets, etc.) can be, if it has to satisfy certain restrictions. Much of extremal combinatorics concerns classes of set systems; this is called extremal set theory. For instance, in an n -element set, what is the largest number of k -element subsets that can pairwise intersect one another? What is the largest number of subsets of which none contains any other? The latter question is answered by Sperner's theorem, which gave rise to much of extremal set theory.

The types of questions addressed in this case are about the largest possible graph which satisfies certain properties. For example, the largest triangle-free graph on 2n vertices is a complete bipartite graph K. Often it is too hard even to find the extremal answer f (n) exactly and one can only give an asymptotic estimate.

Ramsey theory is combinatorics. It states that any sufficiently large configuration will contain some sort of combinatorics. It is an advanced generalization of combinatorics.