Combinatorics

Enumeration focuses on counting the number of certain combinatorial objects, often resulting in finding exact formulas.

Graph Theory studies the properties and applications of graphs, a fundamental set of objects in combinatorics representing pairwise relations.

Design Theory examines combinatorial designs, which are structures with specific properties used in statistical and information-theoretic applications.

Coding Theory deals with the design of error-correcting codes for reliable data transmission over noisy channels.

Combinatorial Optimization is the process of finding an optimal object from a finite set of objects and encompasses algorithms and complexity.

Algebraic Combinatorics involves the use of algebraic techniques and the study of combinatorial structures with algebraic significance.

Topological Combinatorics uses ideas and methods from topology to address combinatorial problems.

Probabilistic Combinatorics applies probability theory to combinatorial problems, allowing for the analysis of random structures.

Combinatorial Number Theory explores number theory problems with a combinatorial approach.

Extremal Combinatorics studies extremal properties of combinatorial structures, asking 'how large or how small' a collection of objects can be if it must satisfy certain restrictions.

Order Theory is interested in the theory of order and relations within sets.

Matroid Theory examines and generalizes the notion of linear independence from vector spaces to arbitrary sets.

Incidence Geometry focuses on the study of geometric structures based on their incidence properties.

Combinatorial Geometry deals with configurations of points, lines, planes, and other geometric figures with respect to combinatorial properties.

Infinitary Combinatorics extends the study of combinatorial principles to infinite sets, including infinite graph theory and Ramsey theory.

Partition Theory studies the ways a number or a set can be decomposed into subsets with specific properties.