In mathematics, a semigroup is an algebraic structure consisting of a set together with an associative internal binary operation on it.
Ramsey theory, named after the British mathematician and philosopher Frank P. Ramsey, is a branch of the mathematical field of combinatorics 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?"
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 r and s be any two positive integers. Ramsey's theorem states that there exists a least positive integer R(r, s) for which every blue-red edge colouring of the complete graph on R(r, s) vertices contains a blue clique on r vertices or a red clique on s vertices. (Here R(r, s) signifies an integer that depends on both r and s.)
In abstract algebra, the free monoid on a set is the monoid whose elements are all the finite sequences of zero or more elements from that set, with string concatenation as the monoid operation and with the unique sequence of zero elements, often called the empty string and denoted by ε or λ, as the identity element. The free monoid on a set A is usually denoted A∗. The free semigroup on A is the subsemigroup of A∗ containing all elements except the empty string. It is usually denoted A+.
In arithmetic combinatorics, Szemerédi's theorem is a result concerning arithmetic progressions in subsets of the integers. In 1936, Erdős and Turán conjectured that every set of integers A with positive natural density contains a k-term arithmetic progression for every k. Endre Szemerédi proved the conjecture in 1975.
In mathematics, the Hales–Jewett theorem is a fundamental combinatorial result of Ramsey theory named after Alfred W. Hales and Robert I. Jewett, concerning the degree to which high-dimensional objects must necessarily exhibit some combinatorial structure; it is impossible for such objects to be "completely random".
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.
In mathematics, a syndetic set is a subset of the natural numbers having the property of "bounded gaps": that the sizes of the gaps in the sequence of natural numbers is bounded.
In mathematics, piecewise syndeticity is a notion of largeness of subsets of the natural numbers.
In mathematics, a thick set is a set of integers that contains arbitrarily long intervals. That is, given a thick set , for every , there is some such that .
In combinatorics, a branch of mathematics, partition regularity is one notion of largeness for a collection of sets.
In mathematics, the Milliken–Taylor theorem in combinatorics is a generalization of both Ramsey's theorem and Hindman's theorem. It is named after Keith Milliken and Alan D. Taylor.
Hillel "Harry" Furstenberg is a German-born American-Israeli mathematician and professor emeritus at the Hebrew University of Jerusalem. He is a member of the Israel Academy of Sciences and Humanities and U.S. National Academy of Sciences and a laureate of the Abel Prize and the Wolf Prize in Mathematics. He is known for his application of probability theory and ergodic theory methods to other areas of mathematics, including number theory and Lie groups.
In mathematics, infinitary combinatorics, or combinatorial set theory, is an extension of ideas in combinatorics to infinite sets. Some of the things studied include continuous graphs and trees, extensions of Ramsey's theorem, and Martin's axiom. Recent developments concern combinatorics of the continuum and combinatorics on successors of singular cardinals.
Ergodic Ramsey theory is a branch of mathematics where problems motivated by additive combinatorics are proven using ergodic theory.
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.
In mathematics, structural Ramsey theory is a categorical generalisation of Ramsey theory, rooted in the idea that many important results of Ramsey theory have "similar" logical structures. The key observation is noting that these Ramsey-type theorems can be expressed as the assertion that a certain category has the Ramsey property.
Thomas Craig Brown is an American-Canadian mathematician, Ramsey Theorist, and Professor Emeritus at Simon Fraser University.
Neil Hindman is an American mathematician and Professor Emeritus at Howard University. His research focuses on various areas within mathematics, including topology, Stone-Čech compactification, discrete systems, and Ramsey theory.
