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Rado's Theorem (Ramsey Theory)
Rado's theorem is a theorem from the branch of mathematics known as Ramsey theory. It is named for the German mathematician Richard Rado. It was proved in his thesis, ''Studien zur Kombinatorik''. Statement Let A \mathbf = \mathbf be a system of linear equations, where A is a matrix with integer entries. This system is said to be r''-regular'' if, for every r-coloring of the natural numbers 1, 2, 3, ..., the system has a monochromatic solution. A system is ''regular'' if it is ''r-regular'' for all ''r'' ≥ 1. Rado's theorem states that a system A \mathbf = \mathbf is regular if and only if the matrix ''A'' satisfies the ''columns condition''. Let ''ci'' denote the ''i''-th column of ''A''. The matrix ''A'' satisfies the columns condition provided that there exists a partition ''C''1, ''C''2, ..., ''C''''n'' of the column indices such that if s_i = \Sigma_c_j, then # ''s''1 = 0 # for all ''i'' ≥ 2, ''si'' can be written as a ration ... [...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 ... [...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 in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Richard Rado
Richard Rado FRS (28 April 1906 – 23 December 1989) was a German-born British mathematician whose research concerned combinatorics and graph theory. He was Jewish and left Germany to escape Nazi persecution. He earned two PhDs: in 1933 from the University of Berlin, and in 1935 from the University of Cambridge. He was interviewed in Berlin by Lord Cherwell for a scholarship given by the chemist Sir Robert Mond which provided financial support to study at Cambridge. After he was awarded the scholarship, Rado and his wife left for the UK in 1933. He was appointed Professor of Mathematics at the University of Reading in 1954 and remained there until he retired in 1971. Contributions Rado made contributions in combinatorics and graph theory including 18 papers with Paul Erdős. In graph theory, the Rado graph, a countably infinite graph containing all countably infinite graphs as induced subgraphs, is named after Rado. He rediscovered it in 1964 after previous works on the same g ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Béla Bollobás
Béla Bollobás FRS (born 3 August 1943) is a Hungarian-born British mathematician who has worked in various areas of mathematics, including functional analysis, combinatorics, graph theory, and percolation. He was strongly influenced by Paul Erdős since the age of 14. Early life and education As a student, he took part in the first three International Mathematical Olympiads, winning two gold medals. Paul Erdős invited Bollobás to lunch after hearing about his victories, and they kept in touch afterward. Bollobás' first publication was a joint publication with ErdősBollobás, Béla; Erdös, Paul , Über graphentheoretische Extremalprobleme. (Extremal problems in graph theory.) , Mat. Lapok 13, 143-152 (1962) on extremal problems in graph theory, written when he was in high school in 1962. With Erdős's recommendation to Harold Davenport and a long struggle for permission from the Hungarian authorities, Bollobás was able to spend an undergraduate year in Cambridge, England ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Folkman's Theorem
Folkman's theorem is a theorem in mathematics, and more particularly in arithmetic combinatorics and Ramsey theory. According to this theorem, whenever the natural numbers are partitioned into finitely many subsets, there exist arbitrarily large sets of numbers all of whose sums belong to the same subset of the partition.. The theorem had been discovered and proved independently by several mathematicians,.. before it was named "Folkman's theorem", as a memorial to Jon Folkman, by Graham, Rothschild, and Spencer. Statement of the theorem Let N be the set of positive integers, and suppose that N is partitioned into ''k'' different subsets ''N''1, ''N''2, ... ''N''''k'', where ''k'' is any positive integer. Then Folkman's theorem states that, for every positive integer ''m'', there exists a set ''S''''m'' and an index ''i''''m'' such that ''S''''m'' has ''m'' elements and such that every sum of a nonempty subset of ''S''''m'' belongs to ''N''''i''''m''. Relation to Rado's the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Schur's Theorem
In discrete mathematics, Schur's theorem is any of several theorems of the mathematician Issai Schur. In differential geometry, Schur's theorem is a theorem of Axel Schur. In functional analysis, Schur's theorem is often called Schur's property, also due to Issai Schur. Ramsey theory In Ramsey theory, Schur's theorem states that for any partition of the positive integers into a finite number of parts, one of the parts contains three integers ''x'', ''y'', ''z'' with :x + y = z. For every positive integer ''c'', ''S''(''c'') denotes the smallest number ''S'' such that for every partition of the integers \ into ''c'' parts, one of the parts contains integers ''x'', ''y'', and ''z'' with x + y = z. Schur's theorem ensures that ''S''(''c'') is well-defined for every positive integer ''c''. The numbers of the form ''S''(''c'') are called Schur's number. Folkman's theorem generalizes Schur's theorem by stating that there exist arbitrarily large sets of integers, all of whose ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Van Der Waerden's Theorem
Van der Waerden's theorem is a theorem in the branch of mathematics called Ramsey theory. Van der Waerden's theorem states that for any given positive integers ''r'' and ''k'', there is some number ''N'' such that if the integers are colored, each with one of ''r'' different colors, then there are at least ''k'' integers in arithmetic progression whose elements are of the same color. The least such ''N'' is the Van der Waerden number ''W''(''r'', ''k''), named after the Dutch mathematician B. L. van der Waerden. Example For example, when ''r'' = 2, you have two colors, say and . ''W''(2, 3) is bigger than 8, because you can color the integers from like this: and no three integers of the same color form an arithmetic progression. But you can't add a ninth integer to the end without creating such a progression. If you add a , then the , , and are in arithmetic progression. Alternatively, if you add a , then the , , and are in arithmetic progression. In fact, there is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Subset Sum Problem
The subset sum problem (SSP) is a decision problem in computer science. In its most general formulation, there is a multiset S of integers and a target-sum T, and the question is to decide whether any subset of the integers sum to precisely T''.'' The problem is known to be NP. Moreover, some restricted variants of it are NP-complete too, for example: * The variant in which all inputs are positive. * The variant in which inputs may be positive or negative, and T=0. For example, given the set \, the answer is ''yes'' because the subset \ sums to zero. * The variant in which all inputs are positive, and the target sum is exactly half the sum of all inputs, i.e., T = \frac(a_1+\dots+a_n) . This special case of SSP is known as the partition problem. SSP can also be regarded as an optimization problem: find a subset whose sum is at most ''T'', and subject to that, as close as possible to ''T''. It is NP-hard, but there are several algorithms that can solve it reasonably quickly in pra ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Reduction (complexity)
In computability theory and computational complexity theory, a reduction is an algorithm for transforming one problem into another problem. A sufficiently efficient reduction from one problem to another may be used to show that the second problem is at least as difficult as the first. Intuitively, problem ''A'' is reducible to problem ''B'', if an algorithm for solving problem ''B'' efficiently (if it existed) could also be used as a subroutine to solve problem ''A'' efficiently. When this is true, solving ''A'' cannot be harder than solving ''B''. "Harder" means having a higher estimate of the required computational resources in a given context (e.g., higher time complexity, greater memory requirement, expensive need for extra hardware processor cores for a parallel solution compared to a single-threaded solution, etc.). The existence of a reduction from ''A'' to ''B'', can be written in the shorthand notation ''A'' ≤m ''B'', usually with a subscript on the ≤ to indicate the t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
NP-complete
In computational complexity theory, a problem is NP-complete when: # it is a problem for which the correctness of each solution can be verified quickly (namely, in polynomial time) and a brute-force search algorithm can find a solution by trying all possible solutions. # the problem can be used to simulate every other problem for which we can verify quickly that a solution is correct. In this sense, NP-complete problems are the hardest of the problems to which solutions can be verified quickly. If we could find solutions of some NP-complete problem quickly, we could quickly find the solutions of every other problem to which a given solution can be easily verified. The name "NP-complete" is short for "nondeterministic polynomial-time complete". In this name, "nondeterministic" refers to nondeterministic Turing machines, a way of mathematically formalizing the idea of a brute-force search algorithm. Polynomial time refers to an amount of time that is considered "quick" for a de ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
