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Erdős–Ko–Rado Theorem
In mathematics, the Erdős–Ko–Rado theorem limits the number of Set (mathematics), 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 Erd ... [...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. It simply means the overlapping area of two or more objects or geometries. 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 the original objects. In this approach an intersection can be sometimes undefined, such as for parallel l ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Graph Theory
In mathematics and computer science, graph theory is the study of ''graph (discrete mathematics), graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of ''Vertex (graph theory), vertices'' (also called ''nodes'' or ''points'') which are connected by ''Glossary of graph theory terms#edge, edges'' (also called ''arcs'', ''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 (mathematics), set of vertices (also called nodes or points); * ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fano Plane
In finite geometry, the Fano plane (named 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. In a separate ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complement (set Theory)
In set theory, the complement of a Set (mathematics), set , often denoted by A^c (or ), is the set of Element (mathematics), elements not in . When all elements in the Universe (set theory), universe, i.e. all elements under consideration, are considered to be Element (mathematics), 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 : ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Johnson Graph J(4,2)
Johnson may refer to: People and fictional characters *Johnson (surname), a common surname in English * Johnson (given name), a list of people * List of people with surname Johnson, including fictional characters *Johnson (composer) (1953–2011), Indian film score composer * Johnson (rapper) (born 1979), Danish rapper * Mr. Johnson (born 1966), Nigerian singer Places * Mount Johnson (other) Canada * Johnson, Ontario, township * Johnson (electoral district), provincial electoral district in Quebec * Johnson Point (British Columbia), a headland on the north side of the entrance to Belize Inlet United States * Johnson, Arizona * Johnson, Arkansas, a town * Johnson, Delaware * Johnson, Indiana, an unincorporated town * Johnson, Kentucky * Johnson, Minnesota * Johnson, Nebraska * Johnson, New York * Johnson, Ohio, an unincorporated community * Johnson, Oklahoma * Johnson, Utah * Johnson, Vermont, a town ** Johnson (village), Vermont * Johnson, Washington * Johnso ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dilworth's Theorem
In mathematics, in the areas of order theory and combinatorics, Dilworth's theorem states that, in any finite partially ordered set, the maximum size of an antichain of incomparable elements equals the minimum number of chains needed to cover all elements. This number is called the width of the partial order. The theorem is named for the mathematician Robert P. Dilworth, who published it in 1950. A version of the theorem for infinite partially ordered sets states that, when there exists a decomposition into finitely many chains, or when there exists a finite upper bound on the size of an antichain, the sizes of the largest antichain and of the smallest chain decomposition are again equal. Statement 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 st ... [...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. In each case, the theorem gives a necessity and sufficiency, necessary and sufficient condition for an object to exist: * The Combinatorics, combinatorial formulation answers whether a Finite set, finite collection of Set (mathematics), sets has a transversal (combinatorics), transversal—that is, whether an element can be chosen from each set without repetition. Hall's condition is that for any group of sets from the collection, the total unique elements they contain is at least as large as the number of sets in the group. * The Graph theory, graph theoretic formulation answers whether a finite bipartite graph has a perfect matching—that is, a way to match each vertex from one group uniquely to an adjacent vertex from the other group. Hall's condition is that any subset of vertices from one group has a neighbourhood (graph theory), neighbourhood of equal or greater size. Combinat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Louis J
Louis may refer to: People * Louis (given name), origin and several individuals with this name * Louis (surname) * Louis (singer), Serbian singer Other uses * Louis (coin), a French coin * HMS Louis, HMS ''Louis'', two ships of the Royal Navy See also * Derived terms * King Louis (other) * Saint Louis (other) * Louis Cruise Lines * Louis dressing, for salad * Louis Quinze, design style Associated terms * Lewis (other) * Louie (other) * Luis (other) * Louise (other) * Louisville (other) 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
The University of Manchester is a public university, public research university in Manchester, England. The main campus is south of Manchester city centre, Manchester City Centre on Wilmslow Road, Oxford Road. The University of Manchester is considered a red brick university, a product of the civic university movement of the late 19th century. The current University of Manchester was formed in 2004 following the merger of the University of Manchester Institute of Science and Technology (UMIST) and the Victoria University of Manchester. This followed a century of the two institutions working closely with one another. Additionally, the university owns and operates major cultural assets such as the Manchester Museum, The Whitworth art gallery, the John Rylands Library, the Tabley House, Tabley House Collection and the Jodrell Bank Observatory – a UNESCO World Heritage Site. The University of Manchester Institute of Science and Technology had its origins in the Manchester Mechan ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
University Of Cambridge
The University of Cambridge is a Public university, public collegiate university, collegiate research university in Cambridge, England. Founded in 1209, the University of Cambridge is the List of oldest universities in continuous operation, world's third-oldest university in continuous operation. The university's founding followed the arrival of scholars who left the University of Oxford for Cambridge after a dispute with local townspeople. The two ancient university, ancient English universities, although sometimes described as rivals, share many common features and are often jointly referred to as Oxbridge. In 1231, 22 years after its founding, the university was recognised with a royal charter, granted by Henry III of England, King Henry III. The University of Cambridge includes colleges of the University of Cambridge, 31 semi-autonomous constituent colleges and List of institutions of the University of Cambridge#Schools, Faculties, and Departments, over 150 academic departm ... [...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 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 Sichuan University
Sichuan University (SCU) is a public university in Chengdu, Sichuan, C ...
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Fractional Chromatic Number
Fractional coloring is a topic in a 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 coloring out ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
