Extremal combinatorics

Extremal combinatorics - a field of combinatorial mathematics which has undergone a period of spectacular growth in recent decades. The word "extremal" comes from the nature of problems this field deals with: if a collection of finite objects (numbers, graphs, vectors, sets, etc.) satisfies certain restrictions, how large or how small can it be?

For example, how many people can we invite to a party where among each three people there are two who know each other and two who don't know each other? An easy Ramsey-type argument shows that at most five persons can attend such a party. Or, suppose we are given a finite set of nonzero integers, and are asked to mark an as large as possible subset of them under the restriction that the sum of any two marked integers cannot be marked. It appears that (independent of what the given integers actually are!) we can always mark at least one-third of them.

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