In combinatorics, theorems refer to established mathematical statements that have been proven based on axioms and previously established theorems. Combinatorics itself is the branch of mathematics dealing with the counting, arrangement, and combination of objects. It often involves discrete structures and discrete quantities.
Baranyai's theorem is a result in combinatorial design theory, specifically within the area of finite set systems. It is named after its discoverer, Zsolt Baranyai. The theorem deals with the partitioning of a complete graph into smaller structures, namely, it provides conditions under which it is possible to partition the complete graph on a certain number of vertices into disjoint complete subgraphs of smaller sizes.
Dilworth's theorem is a result in order theory, a branch of mathematics that studies the properties of ordered sets. The theorem states that in any finite partially ordered set (poset), the size of the largest antichain (a subset of elements in which no two elements are comparable) is equal to the smallest number of chains (totally ordered subsets) that can cover the poset. In more formal terms: - Let \( P \) be a finite poset.
The Erdős–Fuchs theorem is a result in number theory concerning the distribution of prime numbers. It provides a characterization of the divisibility of numbers in the context of prime factors.
The Erdős–Rado theorem is a result in combinatorial set theory that deals with families of sets and their intersections. It is named after mathematicians Paul Erdős and Richard Rado, who developed the theorem in the context of infinite combinatorics.
The Erdős–Tetali theorem is a result in combinatorial mathematics related to the study of extremal graph theory. Specifically, it deals with the relationship between the number of edges in a graph and the degrees of its vertices.
Hall's Marriage Theorem is a result in combinatorial mathematics, specifically in the area of graph theory and bipartite matching. It provides a necessary and sufficient condition for the existence of a perfect matching in a bipartite graph.
The Lagrange inversion theorem is a result in combinatorial mathematics and algebra that provides a formula for finding the coefficients of a power series that is the inverse of another power series. It is particularly useful when dealing with formal power series and can be applied in various areas including combinatorics, algebraic geometry, and differential equations.
Mirsky's theorem is a result in the field of linear algebra and matrix theory that pertains to the relationship between the rank of a matrix and the ranks of its associated matrices.
Mnëv's universality theorem is a result in the field of mathematical logic and combinatorial geometry, specifically relating to the arrangement and properties of arrangements of points in the projective plane. It asserts that certain geometric configurations can be used to describe and encode a broad class of mathematical structures. The theorem indicates that the space of geometric configurations — particularly those involving points and lines in a projective space — is rich enough to capture the complexity of various combinatorial and algebraic structures.
The Szemerédi–Trotter theorem is a fundamental result in combinatorial geometry that provides bounds on the incidences between points and lines in the plane. Specifically, it addresses how many points lie on a set of lines, providing a relationship between three parameters: the number of points, the number of lines, and the number of incidences (that is, points that lie on those lines).
Articles by others on the same topic
There are currently no matching articles.