Combinatorics and Graph Theory Concepts

A collection of vocabulary terms and definitions related to combinatorics, graph theory, and probability.

Combinatorics

The branch of mathematics dealing with combinations and arrangements of objects.

Pigeonhole Principle

If n items are put into m containers and n > m, then at least one container must contain more than one item.

Inclusion-Exclusion Principle

A formula to calculate the size of the union of multiple sets by including the sizes of the sets and excluding the sizes of their intersections.

K-subsets

Subsets of size k from a set of size n.

Antichain

A collection of sets where no one set is contained within another.

Intersecting Family of Sets

A family of sets is intersecting if the intersection of any two sets in the family is not empty.

Erdos-Szekeres Theorem

A theorem stating that in a sequence of n distinct real numbers, there is either an increasing subsequence of length s+1 or a decreasing subsequence of length t+1.

Strongly Regular Graph

A graph that is both regular (every vertex has the same number of neighbors) and meets specific conditions regarding common neighbors between adjacent and non-adjacent vertices.

Random Event

In probability theory, an event that can be described with a subset of the sample space.

Laplacian Matrix

A matrix representation of a graph that encodes the adjacencies and degrees of the vertices.

Ramsey Theorem

A theorem in combinatorial mathematics that asserts for any given integers, there exists a minimum number of vertices such that a complete graph with that number of vertices will have a complete subgraph of a specified size.

Error-Correcting Code

A method used to control errors in data transmission or storage, allowing the detection and correction of errors that occur during transmission.