Extremal combinatorics is a field of combinatorics, which is itself a part of mathematics. 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. Combinatorics is a branch of mathematics that studies collections (usually finite) of objects that satisfy specified criteria. ... Mathematics is often defined as the study of topics such as quantity, structure, space, and change. ... Number is the current mathematics collaboration of the week! Please help improve it to featured article standard. ... A vector space (or linear space) is the basic object of study in the branch of mathematics called linear algebra. ... In mathematics, a set can be thought of as any collection of distinct things considered as a whole. ...
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. Ramsey theory, named for Frank P. Ramsey, is a branch of mathematics that studies the conditions under which order must appear. ...
References
Stasys Jukna, Extremal Combinatorics, With Applications in Computer Science (preface). Springer-Verlag, 2001. ISBN 3-540-66313-4.
Combinatorics is a branch of mathematics that studies collections (usually finite) of objects that satisfy specified criteria.
Combinatorics is as much about problem solving as theory building, though it has developed powerful theoretical methods, especially since the later twentieth century.
Modern combinatorics began to develop in the late nineteenth century and became a distinguishable field of study in the twentieth century, partly through the publication of the systematic enumerative treatise Combinatory Analysis by Percy Alexander MacMahon in 1915 and the work of R.A. Fisher in design of experiments in the 1920s.
Much of combinatorics is about graphs, to whose study all types of combinatorics can contribute.
Enumerative combinatorics came to prominence because counting configurations is essential to elementary probability, starting with the work of Pascal and others.
Two of the most prominent combinatorialists of recent times were the prolific problem-raiser and problem-solver Paul Erdős, who worked mainly on extremal questions, and Gian-Carlo Rota, who helped to formalize the subject beginning in the 1960s, mostly in enumeration and algebraization.
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