Combinatorics is a somewhat paradoxical area of mathematics, being at once simple and complex, poor and rich, easy and difficult, pure and applied. To be more precise, there are problems that are simple to state, but have complex solutions ; we use weak hypotheses, but the consequences are often surprisingly rich ; demonstrations can be short and easy to understand, but ingenious and difficult to discover ; and the objects we study, such as graphs or families of subsets of a finite set, are of purely mathematical interest, but the results concerning them have applications in many other fields, such as computer science, economics or epidemiology.

One of the most important subfields of Combinatorics is graph theory. A graph is a collection of vertices, some of which are linked by edges. Graphs can be used to represent a wide range of real-world phenomena. For example, vertices could represent websites, and edges, the links between sites. Or vertices could represent people, and edges the potential transmissions of the Covid. In general, graphs are abstract representations of networks : vertices represent the objects in the network, and edges represent the relationships between these objects.

Another subfield, additive Combinatorics, concerns the additive structure of sets of integers and the additive relations between their elements. One of the highlights of this subfield is a famous theorem by mathematician Endre Szemerédi, according to which any dense set of integers contains arithmetic progressions of any length.