Extremal combinatorics

Extremal combinatorics is a field of combinatorics, which is itself a part of mathematics. Extremale combinatorics study how large or how small collections of finite objects (numbers, graphs, vectors, sets, etc.) can be, if they have to satisfy certain restrictions.

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 do not? 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 subset as large as possible 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.

References

• Stasys Jukna, Extremal Combinatorics, With Applications in Computer Science (preface). Springer-Verlag, 2001. ISBN 3-540-66313-4.