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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 th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Intersecting Set Families 2-of-4
In mathematics, the intersection of two or more objects is another object consisting of everything that is contained in all of the objects simultaneously. For example, in Euclidean geometry, when two lines in a plane are not parallel, their intersection is the point at which they meet. More generally, in set theory, the intersection of sets is defined to be the set of elements which belong to all of them. Unlike the Euclidean definition, this does not presume that the objects under consideration lie in a common space. Intersection is one of the basic concepts of geometry. An intersection can have various geometric shapes, but a point is the most common in a plane geometry. Incidence geometry defines an intersection (usually, of flats) as an object of lower dimension that is incident to each of original objects. In this approach an intersection can be sometimes undefined, such as for parallel lines. In both cases the concept of intersection relies on logical conju ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Graph Theory
In mathematics, graph theory is the study of ''graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of '' vertices'' (also called ''nodes'' or ''points'') which are connected by '' edges'' (also called ''links'' or ''lines''). A distinction is made between undirected graphs, where edges link two vertices symmetrically, and directed graphs, where edges link two vertices asymmetrically. Graphs are one of the principal objects of study in discrete mathematics. Definitions Definitions in graph theory vary. The following are some of the more basic ways of defining graphs and related mathematical structures. Graph In one restricted but very common sense of the term, a graph is an ordered pair G=(V,E) comprising: * V, a set of vertices (also called nodes or points); * E \subseteq \, a set of edges (also called links or lines), which are unordered pairs of vertices (that is, an edge is associated with t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fano Plane
In finite geometry, the Fano plane (after Gino Fano) is a finite projective plane with the smallest possible number of points and lines: 7 points and 7 lines, with 3 points on every line and 3 lines through every point. These points and lines cannot exist with this pattern of incidences in Euclidean geometry, but they can be given coordinates using the finite field with two elements. The standard notation for this plane, as a member of a family of projective spaces, is . Here stands for "projective geometry", the first parameter is the geometric dimension (it is a plane, of dimension 2) and the second parameter is the order (the number of points per line, minus one). The Fano plane is an example of a finite incidence structure, so many of its properties can be established using combinatorial techniques and other tools used in the study of incidence geometries. Since it is a projective space, algebraic techniques can also be effective tools in its study. Homogeneous coordinat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complement (set Theory)
In set theory, the complement of a set , often denoted by (or ), is the set of elements not in . When all sets in the universe, i.e. all sets under consideration, are considered to be members of a given set , the absolute complement of is the set of elements in that are not in . The relative complement of with respect to a set , also termed the set difference of and , written B \setminus A, is the set of elements in that are not in . Absolute complement Definition If is a set, then the absolute complement of (or simply the complement of ) is the set of elements not in (within a larger set that is implicitly defined). In other words, let be a set that contains all the elements under study; if there is no need to mention , either because it has been previously specified, or it is obvious and unique, then the absolute complement of is the relative complement of in : A^\complement = U \setminus A. Or formally: A^\complement = \. The absolute complement of is u ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Johnson Graph J(4,2)
Johnson is a surname of Anglo-Norman origin meaning "Son of John". It is the second most common in the United States and 154th most common in the world. As a common family name in Scotland, Johnson is occasionally a variation of ''Johnston'', a habitational name. Etymology The name itself is a patronym of the given name '' John'', literally meaning "son of John". The name ''John'' derives from Latin '' Johannes'', which is derived through Greek ''Iōannēs'' from Hebrew '' Yohanan'', meaning "Yahweh has favoured". Origin The name has been extremely popular in Europe since the Christian era as a result of it being given to St John the Baptist, St John the Evangelist and nearly one thousand other Christian saints. Other Germanic languages * Swedish: Johnsson, Jonsson * Icelandic: Jónsson See also * List of people with surname Johnson * Gjoni (Gjonaj) * Ioannou * Jensen *Johansson * Johns * Johnsson * Johnston *Johnstone *Jones * Jonson *Jonsson Jonsson is a surnam ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dilworth's Theorem
In mathematics, in the areas of order theory and combinatorics, Dilworth's theorem characterizes the width of any finite partially ordered set in terms of a partition of the order into a minimum number of chains. It is named for the mathematician . An antichain in a partially ordered set is a set of elements no two of which are comparable to each other, and a chain is a set of elements every two of which are comparable. A chain decomposition is a partition of the elements of the order into disjoint chains. Dilworth's theorem states that, in any finite partially ordered set, the largest antichain has the same size as the smallest chain decomposition. Here, the size of the antichain is its number of elements, and the size of the chain decomposition is its number of chains. The width of the partial order is defined as the common size of the antichain and chain decomposition. A version of the theorem for infinite partially ordered sets states that, when there exists a decomposition i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hall's Marriage Theorem
In mathematics, Hall's marriage theorem, proved by , is a theorem with two equivalent formulations: * The combinatorial formulation deals with a collection of finite sets. It gives a necessary and sufficient condition for being able to select a distinct element from each set. * The graph theoretic formulation deals with a bipartite graph. It gives a necessary and sufficient condition for finding a matching that covers at least one side of the graph. Combinatorial formulation Statement Let \mathcal F be a family of finite sets. Here, \mathcal F is itself allowed to be infinite (although the sets in it are not) and to contain the same set multiple times. Let X be the union of all the sets in \mathcal F, the set of elements that belong to at least one of its sets. A transversal for F is a subset of X that can be obtained by choosing a distinct element from each set in \mathcal F. This concept can be formalized by defining a transversal to be the image of an injective function f: ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Louis J
Louis may refer to: * Louis (coin) * Louis (given name), origin and several individuals with this name * Louis (surname) * Louis (singer), Serbian singer * HMS Louis, HMS ''Louis'', two ships of the Royal Navy See also Derived or associated terms * Lewis (other) * Louie (other) * Luis (other) * Louise (other) * Louisville (other) * Louis Cruise Lines * Louis dressing, for salad * Louis Quinze, design style Associated names * * Chlodwig, the origin of the name Ludwig, which is translated to English as "Louis" * Ladislav and László - names sometimes erroneously associated with "Louis" * Ludovic, Ludwig (other), Ludwig, Ludwick, Ludwik, names sometimes translated to English as "Louis" {{disambiguation ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
University Of Manchester
, mottoeng = Knowledge, Wisdom, Humanity , established = 2004 – University of Manchester Predecessor institutions: 1956 – UMIST (as university college; university 1994) 1904 – Victoria University of Manchester 1880 – Victoria University 1851 – Owens College 1824 – Manchester Mechanics' Institute , endowment = £242.2 million (2021) , budget = £1.10 billion (2020–21) , chancellor = Nazir Afzal (from August 2022) , head_label = President and vice-chancellor , head = Nancy Rothwell , academic_staff = 5,150 (2020) , total_staff = 12,920 (2021) , students = 40,485 (2021) , undergrad = () , postgrad = () , city = Manchester , country = England, United Kingdom , campus = Urban and suburban , colours = Manchester Purple Manchester Yellow , free_label = Scarf , free = , website = , logo = UniOfManchesterLogo.svg , affiliations = Universities Research Association Sutton 30 Russell Group EUA N8 Group NWUA ACUUniversities UK The Universit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
University Of Cambridge
, mottoeng = Literal: From here, light and sacred draughts. Non literal: From this place, we gain enlightenment and precious knowledge. , established = , other_name = The Chancellor, Masters and Scholars of the University of Cambridge , type = Public research university , endowment = £7.121 billion (including colleges) , budget = £2.308 billion (excluding colleges) , chancellor = The Lord Sainsbury of Turville , vice_chancellor = Anthony Freeling , students = 24,450 (2020) , undergrad = 12,850 (2020) , postgrad = 11,600 (2020) , city = Cambridge , country = England , campus_type = , sporting_affiliations = The Sporting Blue , colours = Cambridge Blue , website = , logo = University of Cambridge logo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ke Zhao
Ke Zhao or Chao Ko (, April 12, 1910 – November 8, 2002) was a Chinese mathematician born in Wenling, Taizhou, Zhejiang. Biography Ke graduated from Tsinghua University in 1933 and obtained his doctorate from the University of Manchester under Louis Mordell in 1937. His main fields of study were algebra, number theory and combinatorics. Some of his major contributions included his work on quadratic forms, the 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 ... and his theorem on Catalan's conjecture. In 1955, he was one of the founding members of the Chinese Academy of Sciences. He was later a professor at Sichuan University [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fractional Chromatic Number
Fractional coloring is a topic in a young branch of graph theory known as fractional graph theory. It is a generalization of ordinary graph coloring. In a traditional graph coloring, each vertex in a graph is assigned some color, and adjacent vertices — those connected by edges — must be assigned different colors. In a fractional coloring however, a ''set'' of colors is assigned to each vertex of a graph. The requirement about adjacent vertices still holds, so if two vertices are joined by an edge, they must have no colors in common. Fractional graph coloring can be viewed as the linear programming relaxation of traditional graph coloring. Indeed, fractional coloring problems are much more amenable to a linear programming approach than traditional coloring problems. Definitions A ''b''-fold coloring of a graph ''G'' is an assignment of sets of size ''b'' to vertices of a graph such that adjacent vertices receive disjoint sets. An ''a'':''b''-coloring is a ''b''-fold colorin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
