Monotone class theorem
In measure theory and probability, the monotone class theorem connects monotone classes and sigma-algebras. The theorem says that the smallest containing an algebra of sets is precisely the smallest 𝜎
Sperner's theorem
Sperner's theorem, in discrete mathematics, describes the largest possible families of finite sets none of which contain any other sets in the family. It is one of the central results in extremal set
Finite character
In mathematics, a family of sets is of finite character if for each , belongs to if and only if every finite subset of belongs to . That is, 1.
* For each , every finite subset of belongs to . 2.
*
Near polygon
In mathematics, a near polygon is an incidence geometry introduced by Ernest E. Shult and Arthur Yanushka in 1980. Shult and Yanushka showed the connection between the so-called tetrahedrally closed l
Erdős–Ko–Rado theorem
In mathematics, the Erdős–Ko–Rado theorem limits the number of sets in a family of sets for which every two sets have at least one element in common. Paul Erdős, Chao Ko, and Richard Rado proved the t
Generalized quadrangle
In geometry, a generalized quadrangle is an incidence structure whose main feature is the lack of any triangles (yet containing many quadrangles). A generalized quadrangle is by definition a polar spa
Maximum coverage problem
The maximum coverage problem is a classical question in computer science, computational complexity theory, and operations research.It is a problem that is widely taught in approximation algorithms. As
Ring of sets
In mathematics, there are two different notions of a ring of sets, both referring to certain families of sets. In order theory, a nonempty family of sets is called a ring (of sets) if it is closed und
Sauer–Shelah lemma
In combinatorial mathematics and extremal set theory, the Sauer–Shelah lemma states that every family of sets with small VC dimension consists of a small number of sets. It is named after and Saharon
Ultrafilter (set theory)
In the mathematical field of set theory, an ultrafilter is a maximal proper filter: it is a filter on a given non-empty set which is a certain type of non-empty family of subsets of that is not equal
Pi-system
In mathematics, a π-system (or pi-system) on a set is a collection of certain subsets of such that
* is non-empty.
* If then That is, is a non-empty family of subsets of that is closed under non-emp
Σ-Algebra of τ-past
The σ-algebra of τ-past, (also named stopped σ-algebra, stopped σ-field, or σ-field of τ-past) is a σ-algebra associated with a stopping time in the theory of stochastic processes, a branch of probabi
Property B
In mathematics, Property B is a certain set theoretic property. Formally, given a finite set X, a collection C of subsets of X has Property B if we can partition X into two disjoint subsets Y and Z su
Kruskal–Katona theorem
In algebraic combinatorics, the Kruskal–Katona theorem gives a complete characterization of the f-vectors of abstract simplicial complexes. It includes as a special case the Erdős–Ko–Rado theorem and
Abstract simplicial complex
In combinatorics, an abstract simplicial complex (ASC), often called an abstract complex or just a complex, is a family of sets that is closed under taking subsets, i.e., every subset of a set in the
Laminar set family
In combinatorics, a laminar set family is a set family in which each pair of sets are either disjoint or related by containment. Formally, a set family {S1, S2, ...} is called laminar if for every i,
Teichmüller–Tukey lemma
In mathematics, the Teichmüller–Tukey lemma (sometimes named just Tukey's lemma), named after John Tukey and Oswald Teichmüller, is a lemma that states that every nonempty collection of finite charact
Disjoint sets
In mathematics, two sets are said to be disjoint sets if they have no element in common. Equivalently, two disjoint sets are sets whose intersection is the empty set. For example, {1, 2, 3} and {4, 5,
Steiner system
In combinatorial mathematics, a Steiner system (named after Jakob Steiner) is a type of block design, specifically a t-design with λ = 1 and t = 2 or (recently) t ≥ 2. A Steiner system with parameters
Nerve complex
In topology, the nerve complex of a set family is an abstract complex that records the pattern of intersections between the sets in the family. It was introduced by Pavel Alexandrov and now has many v
Dedekind number
In mathematics, the Dedekind numbers are a rapidly growing sequence of integers named after Richard Dedekind, who defined them in 1897. The Dedekind number M(n) counts the number of monotone boolean f
Delta-ring
In mathematics, a non-empty collection of sets is called a δ-ring (pronounced "delta-ring") if it is closed under union, relative complementation, and countable intersection. The name "delta-ring" ori
Dynkin system
A Dynkin system, named after Eugene Dynkin is a collection of subsets of another universal set satisfying a set of axioms weaker than those of 𝜎-algebra. Dynkin systems are sometimes referred to as 𝜆-
Field of sets
In mathematics, a field of sets is a mathematical structure consisting of a pair consisting of a set and a family of subsets of called an algebra over that contains the empty set as an element, and is
Noncrossing partition
In combinatorial mathematics, the topic of noncrossing partitions has assumed some importance because of (among other things) its application to the theory of free probability. The number of noncrossi
Ultrafilter
In the mathematical field of order theory, an ultrafilter on a given partially ordered set (or "poset") is a certain subset of namely a maximal filter on that is, a proper filter on that cannot be enl
Union-closed sets conjecture
In combinatorics, the union-closed sets conjecture is a problem, posed by Péter Frankl in 1979 and is still open. A family of sets is said to be union-closed if the union of any two sets from the fami
Sigma-ring
In mathematics, a nonempty collection of sets is called a 𝜎-ring (pronounced sigma-ring) if it is closed under countable union and relative complementation.
Block design
In combinatorial mathematics, a block design is an incidence structure consisting of a set together with a family of subsets known as blocks, chosen such that frequency of the elements satisfies certa
Partition regularity
In combinatorics, a branch of mathematics, partition regularity is one notion of largeness for a collection of sets. Given a set , a collection of subsets is called partition regular if every set A in
Extremal Problems For Finite Sets
Extremal Problems For Finite Sets is a mathematics book on the extremal combinatorics of finite sets and families of finite sets. It was written by Péter Frankl and Norihide Tokushige, and published i
Set cover problem
The set cover problem is a classical question in combinatorics, computer science, operations research, and complexity theory. It is one of Karp's 21 NP-complete problems shown to be NP-complete in 197
Incidence structure
In mathematics, an incidence structure is an abstract system consisting of two types of objects and a single relationship between these types of objects. Consider the points and lines of the Euclidean
Antimatroid
In mathematics, an antimatroid is a formal system that describes processes in which a set is built up by including elements one at a time, and in which an element, once available for inclusion, remain
Matroid
In combinatorics, a branch of mathematics, a matroid /ˈmeɪtrɔɪd/ is a structure that abstracts and generalizes the notion of linear independence in vector spaces. There are many equivalent ways to def
Helly family
In combinatorics, a Helly family of order k is a family of sets in which every minimal subfamily with an empty intersection has k or fewer sets in it. Equivalently, every finite subfamily such that ev
Sigma-ideal
In mathematics, particularly measure theory, a 𝜎-ideal, or sigma ideal, of a sigma-algebra (𝜎, read "sigma," means countable in this context) is a subset with certain desirable closure properties. It
Universal set
In set theory, a universal set is a set which contains all objects, including itself. In set theory as usually formulated, it can be proven in multiple ways that a universal set does not exist. Howeve
Sperner family
In combinatorics, a Sperner family (or Sperner system; named in honor of Emanuel Sperner), or clutter, is a family F of subsets of a finite set E in which none of the sets contains another. Equivalent
List of set identities and relations
This article lists mathematical properties and laws of sets, involving the set-theoretic operations of union, intersection, and complementation and the relations of set equality and set inclusion. It
Combinatorial design
Combinatorial design theory is the part of combinatorial mathematics that deals with the existence, construction and properties of systems of finite sets whose arrangements satisfy generalized concept
Universe (mathematics)
In mathematics, and particularly in set theory, category theory, type theory, and the foundations of mathematics, a universe is a collection that contains all the entities one wishes to consider in a
Polar space
In mathematics, in the field of geometry, a polar space of rank n (n ≥ 3), or projective index n − 1, consists of a set P, conventionally called the set of points, together with certain subsets of P,
Two-graph
In mathematics, a two-graph is a set of (unordered) triples chosen from a finite vertex set X, such that every (unordered) quadruple from X contains an even number of triples of the two-graph. A regul
Levi graph
In combinatorial mathematics, a Levi graph or incidence graph is a bipartite graph associated with an incidence structure. From a collection of points and lines in an incidence geometry or a projectiv
Partition of a set
In mathematics, a partition of a set is a grouping of its elements into non-empty subsets, in such a way that every element is included in exactly one subset. Every equivalence relation on a set defin
Social golfer problem
In discrete mathematics, the social golfer problem (SGP) is a combinatorial-design problem derived from a question posted in the usenet newsgroup sci.op-research in May 1998. The problem is as follows
Family of sets
In set theory and related branches of mathematics, a collection F of subsets of a given set S is called a family of subsets of S, or a family of sets over S. More generally, a collection of any sets w
Hypergraph
In mathematics, a hypergraph is a generalization of a graph in which an edge can join any number of vertices. In contrast, in an ordinary graph, an edge connects exactly two vertices. Formally, an und
Growth function
The growth function, also called the shatter coefficient or the shattering number, measures the richness of a set family. It is especially used in the context of statistical learning theory, where it
Transversal (combinatorics)
In mathematics, particularly in combinatorics, given a family of sets, here called a collection C, a transversal (also called a cross-section) is a set containing exactly one element from each member
Almost disjoint sets
In mathematics, two sets are almost disjoint if their intersection is small in some sense; different definitions of "small" will result in different definitions of "almost disjoint".
Σ-algebra
In mathematical analysis and in probability theory, a σ-algebra (also σ-field) on a set X is a nonempty collection Σ of subsets of X closed under complement, countable unions, and countable intersecti
Kirkman's schoolgirl problem
Kirkman's schoolgirl problem is a problem in combinatorics proposed by Rev. Thomas Penyngton Kirkman in 1850 as Query VI in The Lady's and Gentleman's Diary (pg.48). The problem states: Fifteen young
Lubell–Yamamoto–Meshalkin inequality
In combinatorial mathematics, the Lubell–Yamamoto–Meshalkin inequality, more commonly known as the LYM inequality, is an inequality on the sizes of sets in a Sperner family, proved by , , , and . It i
Content (measure theory)
In mathematics, a content is a set function that is like a measure, but a content must only be finitely additive, whereas a measure must be countably additive. A content is a real function defined on
Fisher's inequality
Fisher's inequality is a necessary condition for the existence of a balanced incomplete block design, that is, a system of subsets that satisfy certain prescribed conditions in combinatorial mathemati
Greedoid
In combinatorics, a greedoid is a type of set system. It arises from the notion of the matroid, which was originally introduced by Whitney in 1935 to study planar graphs and was later used by Edmonds
Finite intersection property
In general topology, a branch of mathematics, a non-empty family A of subsets of a set is said to have the finite intersection property (FIP) if the intersection over any finite subcollection of is no