'''Extremal combinatorics''' is a field of [[combinatorics]], which is itself a part of [[mathematics]]. ExtremaleExtremal combinatorics studystudies how large or how small collectionsa collection of finite objects ([[number]]s, [[graph (mathematics)|graph]]s, [[vector space|vector]]s, [[set]]s, etc.) can be, if theyit havehas 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 dodon't notknow each other? An easy [[Ramsey theory|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 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.
