Combinatorics is the branch of mathematics studying the enumeration, permutation and combination of sets of elements and the mathematical relations that characterize their properties.
Mathematicians sometimes use the terms "combinatorics" to refer to a large subset of discrete mathematics that include graph theory. In that case, what is called the combinatorics is then referred to as "ENUMERATION".
In proving results in combinatorics, several useful COMBINATORIAL RULES or COMBINATORIAL PRINCIPLES
are commonly recognized and used.
Some of the important principle in this are RAMSEY'S THEOREM, SPERNER THEOREM, VAN DER WAERDEN'S THEOREM, DIRICHLET'S BOX PRINCIPLE etc.
Short reviews of these theorems are:
1) RAMSEY'S THEOREM : Ramsey's theorem is a generalization of DILWORTH'S LEMMA which states for each pair of positive integers and there exists an integer (known as the Ramsey number) such that any graph with nodes contains a clique with at least nodes or an independent set with at least nodes.
2) DIRICHLET'S BOX PRINCIPLE : A.k.a. the pigeonhole principle. Given boxes and objects, at least one box must contain more than one object. This statement has important applications in number theory and was first stated by Dirichlet in 1834. In general, if objects are placed into boxes, then there exists at least one box containing at least objects, where is the ceiling function.