Editing Combinatorics (section)

===Extremal combinatorics===
{{Main|Extremal combinatorics}}Extremal combinatorics studies how large or how small a collection of finite objects ([[number]]s, [[Graph (discrete mathematics)|graphs]], [[Vector space|vectors]], [[Set (mathematics)|sets]], etc.) can be, if it has to satisfy certain restrictions. Much of extremal combinatorics concerns [[Class (set theory)|classes]] of [[set system]]s; this is called extremal set theory. For instance, in an ''n''-element set, what is the largest number of ''k''-element [[subset]]s 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 family#Sperner's theorem|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<sub>n,n</sub>''. Often it is too hard even to find the extremal answer ''f''(''n'') exactly and one can only give an [[asymptotic analysis|asymptotic estimate]].

[[Ramsey theory]] is another part of extremal combinatorics.  It states that any [[sufficiently large]] configuration will contain some sort of order. It is an advanced generalization of the [[pigeonhole principle]].