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Ramsey-Turán Theory
Ramsey-Turán theory is a subfield of extremal graph theory. It studies common generalizations of Ramsey's theorem and Turán's theorem. In brief, Ramsey-Turán theory asks for the maximum number of edges a graph which satisfies constraints on its subgraphs and structure can have. The theory organizes many natural questions which arise in extremal graph theory. The first authors to formalize the central ideas of the theory were Erdős and Sós in 1969, though mathematicians had previously investigated many Ramsey-Turán-type problems. Ramsey's theorem and Turán's theorem Ramsey's theorem for two colors and the complete graph, proved in its original form in 1930, states that for any positive integer there exists an integer large enough that for any coloring of the edges of the complete graph K_n using two colors has a monochoromatic copy of K_k. More generally, for any graphs L_1,\dots,L_r, there is a threshold R=R(L_1,\dots,L_k) such that if n \geq R and the edges of K_n are co ... [...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 co ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ramsey's Theorem
In combinatorics, Ramsey's theorem, in one of its graph-theoretic forms, states that one will find monochromatic cliques in any edge labelling (with colours) of a sufficiently large complete graph. To demonstrate the theorem for two colours (say, blue and red), let and be any two positive integers. Ramsey's theorem states that there exists a least positive integer for which every blue-red edge colouring of the complete graph on vertices contains a blue clique on vertices or a red clique on vertices. (Here signifies an integer that depends on both and .) Ramsey's theorem is a foundational result in combinatorics. The first version of this result was proved by F. P. Ramsey. This initiated the combinatorial theory now called Ramsey theory, that seeks regularity amid disorder: general conditions for the existence of substructures with regular properties. In this application it is a question of the existence of ''monochromatic subsets'', that is, subsets of connected edges ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Turán's Theorem
In graph theory, Turán's theorem bounds the number of edges that can be included in an undirected graph that does not have a complete subgraph of a given size. It is one of the central results of extremal graph theory, an area studying the largest or smallest graphs with given properties, and is a special case of the forbidden subgraph problem on the maximum number of edges in a graph that does not have a given subgraph. An example of an n-vertex graph that does not contain any (r+1)-vertex clique K_ may be formed by partitioning the set of n vertices into r parts of equal or nearly equal size, and connecting two vertices by an edge whenever they belong to two different parts. The resulting graph is the Turán graph T(n,r). Turán's theorem states that the Turán graph has the largest number of edges among all -free -vertex graphs. Turán's theorem, and the Turán graphs giving its extreme case, were first described and studied by Hungarian mathematician Pál Turán in 1941. Th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Graph (discrete Mathematics)
In discrete mathematics, and more specifically in graph theory, a graph is a structure amounting to a set of objects in which some pairs of the objects are in some sense "related". The objects correspond to mathematical abstractions called '' vertices'' (also called ''nodes'' or ''points'') and each of the related pairs of vertices is called an ''edge'' (also called ''link'' or ''line''). Typically, a graph is depicted in diagrammatic form as a set of dots or circles for the vertices, joined by lines or curves for the edges. Graphs are one of the objects of study in discrete mathematics. The edges may be directed or undirected. For example, if the vertices represent people at a party, and there is an edge between two people if they shake hands, then this graph is undirected because any person ''A'' can shake hands with a person ''B'' only if ''B'' also shakes hands with ''A''. In contrast, if an edge from a person ''A'' to a person ''B'' means that ''A'' owes money to ''B'', th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Paul Erdős
Paul Erdős ( hu, Erdős Pál ; 26 March 1913 – 20 September 1996) was a Hungarian mathematician. He was one of the most prolific mathematicians and producers of mathematical conjectures of the 20th century. pursued and proposed problems in discrete mathematics, graph theory, number theory, mathematical analysis, approximation theory, set theory, and probability theory. Much of his work centered around discrete mathematics, cracking many previously unsolved problems in the field. He championed and contributed to Ramsey theory, which studies the conditions in which order necessarily appears. Overall, his work leaned towards solving previously open problems, rather than developing or exploring new areas of mathematics. Erdős published around 1,500 mathematical papers during his lifetime, a figure that remains unsurpassed. He firmly believed mathematics to be a social activity, living an itinerant lifestyle with the sole purpose of writing mathematical papers with other mat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vera T
Vera may refer to: Names *Vera (surname), a surname (including a list of people with the name) * Vera (given name), a given name (including a list of people and fictional characters with the name) **Vera (), archbishop of the archdiocese of Tarragona Places Spain *Vera, Almería, a municipality in the province of Almería, Andalusia * Vera de Bidasoa, a municipality in the autonomous community of Navarra *La Vera, a comarca in the province of Cáceres, Extremadura United States * Vera, Illinois, an unincorporated community * Vera, Kansas, a ghost town *Vera, Missouri, an unincorporated community * Vera, Oklahoma, a town * Vera, Texas, an unincorporated community * Vera, Virginia, an unincorporated community *Veradale, Washington, originally known as Vera, CDP Elsewhere * Vera, Santa Fe, a city in the province of Santa Fe, Argentina * Vera Department, an administrative subdivision (departamento) of the province of Santa Fe * Vera, Mato Grosso, Brazil, a municipality * Cape Ver ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complete Graph
In the mathematical field of graph theory, a complete graph is a simple undirected graph in which every pair of distinct vertices is connected by a unique edge. A complete digraph is a directed graph in which every pair of distinct vertices is connected by a pair of unique edges (one in each direction). Graph theory itself is typically dated as beginning with Leonhard Euler's 1736 work on the Seven Bridges of Königsberg. However, drawings of complete graphs, with their vertices placed on the points of a regular polygon, had already appeared in the 13th century, in the work of Ramon Llull. Such a drawing is sometimes referred to as a mystic rose. Properties The complete graph on vertices is denoted by . Some sources claim that the letter in this notation stands for the German word , but the German name for a complete graph, , does not contain the letter , and other sources state that the notation honors the contributions of Kazimierz Kuratowski to graph theory. has ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Turán Graph
The Turán graph, denoted by T(n,r), is a complete multipartite graph; it is formed by partition of a set, partitioning a set of n vertices into r subsets, with sizes as equal as possible, and then connecting two vertices by an edge if and only if they belong to different subsets. Where q and s are the quotient and remainder of dividing n by r (so n = qr + s), the graph is of the form K_, and the number of edges is : \left(1 - \frac\right)\frac + . The graph has s subsets of size q+ 1 , and r - s subsets of size q; each vertex has degree n-q-1 or n-q. It is a regular graph if n is divisible by r (i.e. when s=0). Turán's theorem Turán graphs are named after Pál Turán, who used them to prove Turán's theorem, an important result in extremal graph theory. By the pigeonhole principle, every set of ''r'' + 1 vertices in the Turán graph includes two vertices in the same partition subset; therefore, the Turán graph does not contain a Clique (graph theory), clique of siz ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Independent Set (graph Theory)
In graph theory, an independent set, stable set, coclique or anticlique is a set of vertices in a graph, no two of which are adjacent. That is, it is a set S of vertices such that for every two vertices in S, there is no edge connecting the two. Equivalently, each edge in the graph has at most one endpoint in S. A set is independent if and only if it is a clique in the graph's complement. The size of an independent set is the number of vertices it contains. Independent sets have also been called "internally stable sets", of which "stable set" is a shortening. A maximal independent set is an independent set that is not a proper subset of any other independent set. A maximum independent set is an independent set of largest possible size for a given graph G. This size is called the independence number of ''G'' and is usually denoted by \alpha(G). The optimization problem of finding such a set is called the maximum independent set problem. It is a strongly NP-hard problem. As suc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
North-Holland
North Holland ( nl, Noord-Holland, ) is a province of the Netherlands in the northwestern part of the country. It is located on the North Sea, north of South Holland and Utrecht, and west of Friesland and Flevoland. In November 2019, it had a population of 2,877,909 and a total area of , of which is water. From the 9th to the 16th century, the area was an integral part of the County of Holland. During this period West Friesland was incorporated. In the 17th and 18th century, the area was part of the province of Holland and commonly known as the Noorderkwartier (English: "Northern Quarter"). In 1840, the province of Holland was split into the two provinces of North Holland and South Holland. In 1855, the Haarlemmermeer was drained and turned into land. The provincial capital is Haarlem (pop. 161,265). The province's largest city and also the largest city in the Netherlands is the Dutch capital Amsterdam, with a population of 862,965 as of November 2019. The King's Commissio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
