Graph theory
Networks of vertices and edges.
26 problemsCombinatorics
Finite structures, arrangements, and counting.
Explore Combinatorics
Discrete geometry
Arrangements, distances, coverings, and extremal geometric configurations.
11 problemsCollections
Browse collectionsGraphs and coloring
Colors, subgraphs, labels and paths that visit every vertex.
20 problemsGeometric extremes
Packing, covering, distances and other limits set by geometry.
11 problemsBases and transversals
Rearranging bases, choosing distinct symbols and listing minimal hitting sets.
4 problemsA few problems to explore
46 totalHadwiger conjecture
A graph that needs many colors is predicted to contain a correspondingly large complete graph after edges are contracted and vertices removed.
Hadwiger–Nelson problem
How many colors are needed to color every point of the plane so that points exactly one unit apart always have different colors?
Hadamard matrix conjecture
Square matrices of plus and minus ones can have mutually orthogonal rows. The conjecture predicts one for every size divisible by four.
Union-closed sets conjecture
If a finite collection of sets contains every union of two of its members, must some element occur in at least half the sets?
Lonely runner conjecture
Runners start together on a circular track at different constant speeds. Each is predicted to eventually be well separated from all the others.
Erdős–Hajnal conjecture
Forbidding one fixed induced graph is predicted to force a large group of vertices that are all connected to each other or all disconnected.