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Extremal Combinatorics
Extremal combinatorics is a field of combinatorics, which is itself a part of mathematics. Extremal combinatorics studies how large or how small a collection of finite objects (numbers, graphs, vectors, sets, etc.) can be, if it has to satisfy certain restrictions. Much of extremal combinatorics concerns classes of sets; this is called extremal set theory. For instance, in an ''n''-element set, what is the largest number of ''k''-element subsets that can pairwise intersect one another? What is the largest number of subsets of which none contains any other? The latter question is answered by Sperner's theorem, which gave rise to much of extremal set theory. Another kind of example: How many people can be invited to a party where among each three people there are two who know each other and two who don't know each other? Ramsey theory shows that at most five persons can attend such a party. Or, suppose we are given a finite set of nonzero integers, and are asked to mark as la ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Combinatorics
Combinatorics is an area of mathematics primarily concerned with counting, both as a means and an end in obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many applications ranging from logic to statistical physics and from evolutionary biology to computer science. Combinatorics is well known for the breadth of the problems it tackles. Combinatorial problems arise in many areas of pure mathematics, notably in algebra, probability theory, topology, and geometry, as well as in its many application areas. Many combinatorial questions have historically been considered in isolation, giving an ''ad hoc'' solution to a problem arising in some mathematical context. In the later twentieth century, however, powerful and general theoretical methods were developed, making combinatorics into an independent branch of mathematics in its own right. One of the oldest and most accessible parts of combinatorics is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ramsey Theory
Ramsey theory, named after the British mathematician and philosopher Frank P. Ramsey, is a branch of mathematics that focuses on the appearance of order in a substructure given a structure of a known size. Problems in Ramsey theory typically ask a question of the form: "how big must some structure be to guarantee that a particular property holds?" More specifically, Ron Graham described Ramsey theory as a "branch of combinatorics". Examples A typical result in Ramsey theory starts with some mathematical structure that is then cut into pieces. How big must the original structure be in order to ensure that at least one of the pieces has a given interesting property? This idea can be defined as partition regularity. For example, consider a complete graph of order ''n''; that is, there are ''n'' vertices and each vertex is connected to every other vertex by an edge. A complete graph of order 3 is called a triangle. Now colour each edge either red or blue. How large must ''n'' be i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Union-closed Sets Conjecture
The union-closed sets conjecture is an open problem in combinatorics posed by Péter Frankl in 1979. A family of sets is said to be ''union-closed'' if the union of any two sets from the family belongs to the family. The conjecture states: For every finite union-closed family of sets, other than the family containing only the empty set, there exists an element that belongs to at least half of the sets in the family. Professor Timothy Gowers has called this "''one of the best known open problems in combinatorics''" and has said that the conjecture "''feels as though it ought to be easy (and as a result has attracted a lot of false proofs over the years). A good way to understand why it isn't easy is to spend an afternoon trying to prove it. That clever averaging argument you had in mind doesn't work ...''" Example The family of sets\varnothing, \, \, \, \consists of five different sets and is union-closed. The element 1 is contained in three of the five sets (and so is the el ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 mathematics. Outlined by Ronald Fisher, a population geneticist and statistician, who was concerned with the design of experiments such as studying the differences among several different varieties of plants, under each of a number of different growing conditions, called ''blocks''. Let: * be the number of varieties of plants; * be the number of blocks. To be a balanced incomplete block design it is required that: * different varieties are in each block, ; no variety occurs twice in any one block; * any two varieties occur together in exactly blocks; * each variety occurs in exactly blocks. Fisher's inequality states simply that :: . Proof Let the incidence matrix be a matrix defined so that is 1 if element is in block and 0 otherwise. Then is a matrix such that and for . Si ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 can be restated in terms of uniform hypergraphs. It is named after Joseph Kruskal and Gyula O. H. Katona, but has been independently discovered by several others. Statement Given two positive integers ''N'' and ''i'', there is a unique way to expand ''N'' as a sum of binomial coefficients as follows: : N=\binom+\binom+\ldots+\binom,\quad n_i > n_ > \ldots > n_j \geq j\geq 1. This expansion can be constructed by applying the greedy algorithm: set ''n''''i'' to be the maximal ''n'' such that N\geq \binom, replace ''N'' with the difference, ''i'' with ''i'' − 1, and repeat until the difference becomes zero. Define : N^=\binom+\binom+\ldots+\binom. Statement for simplicial complexes An integral vector (f_0, f_1, ..., f_) is the ''f''-vector of some (d-1)-d ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 theorem in 1938, but did not publish it until 1961. It is part of the field of combinatorics, and one of the central results of The theorem applies to families of sets that all have the same and are all subsets of some larger set of size One way to construct a family of sets with these parameters, each two sharing an element, is to choose a single element to belong to all the subsets, and then form all of the subsets that contain the chosen element. The Erdős–Ko–Rado theorem states that when n is large enough for the problem to be nontrivial this construction produces the largest possible intersecting families. When n=2r there are other equally-large families, but for larger values of n only the families constructed in this way can be largest. The Erdős–Ko–Rado ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 Norbert Sauer and Saharon Shelah, who published it independently of each other in 1972. The same result was also published slightly earlier and again independently, by Vladimir Vapnik and Alexey Chervonenkis, after whom the VC dimension is named. In his paper containing the lemma, Shelah gives credit also to Micha Perles, and for this reason the lemma has also been called the Perles–Sauer–Shelah lemma.. Buzaglo et al. call this lemma "one of the most fundamental results on VC-dimension", and it has applications in many areas. Sauer's motivation was in the combinatorics of set systems, while Shelah's was in model theory and that of Vapnik and Chervonenkis was in statistics. It has also been applied in discrete geometry. and graph theory.. Definitions and statement If \textstyle \mathcal=\ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Extremal Graph Theory
Extremal graph theory is a branch of combinatorics, itself an area of mathematics, that lies at the intersection of extremal combinatorics and graph theory. In essence, extremal graph theory studies how global properties of a graph influence local substructure. Results in extremal graph theory deal with quantitative connections between various graph properties, both global (such as the number of vertices and edges) and local (such as the existence of specific subgraphs), and problems in extremal graph theory can often be formulated as optimization problems: how big or small a parameter of a graph can be, given some constraints that the graph has to satisfy? A graph that is an optimal solution to such an optimization problem is called an extremal graph, and extremal graphs are important objects of study in extremal graph theory. Extremal graph theory is closely related to fields such as Ramsey theory, spectral graph theory, computational complexity theory, and additive combin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 theory. It is named after Emanuel Sperner, who published it in 1928. This result is sometimes called Sperner's lemma, but the name "Sperner's lemma" also refers to an unrelated result on coloring triangulations. To differentiate the two results, the result on the size of a Sperner family is now more commonly known as Sperner's theorem. Statement A family of sets in which none of the sets is a strict subset of another is called a Sperner family, or an antichain of sets, or a clutter. For example, the family of ''k''-element subsets of an ''n''-element set is a Sperner family. No set in this family can contain any of the others, because a containing set has to be strictly bigger than the set it contains, and in this family all sets have equal size. The value of ''k'' that makes t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Subset
In mathematics, set ''A'' is a subset of a set ''B'' if all elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are unequal, then ''A'' is a proper subset of ''B''. The relationship of one set being a subset of another is called inclusion (or sometimes containment). ''A'' is a subset of ''B'' may also be expressed as ''B'' includes (or contains) ''A'' or ''A'' is included (or contained) in ''B''. A ''k''-subset is a subset with ''k'' elements. The subset relation defines a partial order on sets. In fact, the subsets of a given set form a Boolean algebra under the subset relation, in which the join and meet are given by intersection and union, and the subset relation itself is the Boolean inclusion relation. Definition If ''A'' and ''B'' are sets and every element of ''A'' is also an element of ''B'', then: :*''A'' is a subset of ''B'', denoted by A \subseteq B, or equivalently, :* ''B'' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Class (set Theory)
In set theory and its applications throughout mathematics, a class is a collection of sets (or sometimes other mathematical objects) that can be unambiguously defined by a property that all its members share. Classes act as a way to have set-like collections while differing from sets so as to avoid Russell's paradox (see ). The precise definition of "class" depends on foundational context. In work on Zermelo–Fraenkel set theory, the notion of class is informal, whereas other set theories, such as von Neumann–Bernays–Gödel set theory, axiomatize the notion of "proper class", e.g., as entities that are not members of another entity. A class that is not a set (informally in Zermelo–Fraenkel) is called a proper class, and a class that is a set is sometimes called a small class. For instance, the class of all ordinal numbers, and the class of all sets, are proper classes in many formal systems. In Quine's set-theoretical writing, the phrase "ultimate class" is often used ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
