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Ramsey Class
In the area of mathematics known as Ramsey theory, a Ramsey class is one which satisfies a generalization of Ramsey's theorem. Suppose A, B and C are structures and k is a positive integer. We denote by \binom the set of all subobjects A' of B which are isomorphic to A. We further denote by C \rightarrow (B)^A_k the property that for all partitions X_1 \cup X_2\cup \dots\cup X_k of \binom there exists a B' \in \binom and an 1 \leq i \leq k such that \binom \subseteq X_i. Suppose K is a class of structures closed under isomorphism In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word i ... and substructures. We say the class K has the A-Ramsey property if for ever positive integer k and for every B\in K there is a C \in K such that C \rightarrow (B)^A_k holds. If K has the A-Ramsey property ... [...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] |
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] |
Isomorphism
In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word isomorphism is derived from the Ancient Greek: ἴσος ''isos'' "equal", and μορφή ''morphe'' "form" or "shape". The interest in isomorphisms lies in the fact that two isomorphic objects have the same properties (excluding further information such as additional structure or names of objects). Thus isomorphic structures cannot be distinguished from the point of view of structure only, and may be identified. In mathematical jargon, one says that two objects are . An automorphism is an isomorphism from a structure to itself. An isomorphism between two structures is a canonical isomorphism (a canonical map that is an isomorphism) if there is only one isomorphism between the two structures (as it is the case for solutions of a unive ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Substructure (mathematics)
In mathematical logic, an (induced) substructure or (induced) subalgebra is a structure whose domain is a subset of that of a bigger structure, and whose functions and relations are restricted to the substructure's domain. Some examples of subalgebras are subgroups, submonoids, subrings, subfields, subalgebras of algebras over a field, or induced subgraphs. Shifting the point of view, the larger structure is called an extension or a superstructure of its substructure. In model theory, the term "submodel" is often used as a synonym for substructure, especially when the context suggests a theory of which both structures are models. In the presence of relations (i.e. for structures such as ordered groups or graphs, whose signature is not functional) it may make sense to relax the conditions on a subalgebra so that the relations on a weak substructure (or weak subalgebra) are ''at most'' those induced from the bigger structure. Subgraphs are an example where the distinction ma ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Advances In Mathematics
''Advances in Mathematics'' is a peer-reviewed scientific journal covering research on pure mathematics. It was established in 1961 by Gian-Carlo Rota. The journal publishes 18 issues each year, in three volumes. At the origin, the journal aimed at publishing articles addressed to a broader "mathematical community", and not only to mathematicians in the author's field. Herbert Busemann writes, in the preface of the first issue, "The need for expository articles addressing either all mathematicians or only those in somewhat related fields has long been felt, but little has been done outside of the USSR. The serial publication ''Advances in Mathematics'' was created in response to this demand." Abstracting and indexing The journal is abstracted and indexed in: * [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |