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 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.
Endre Szemerédi is a Hungarian-American mathematician and computer scientist, working in the field of combinatorics and theoretical computer science. He has been the State of New Jersey Professor of computer science at Rutgers University since 1986. He also holds a professor emeritus status at the Alfréd Rényi Institute of Mathematics of the Hungarian Academy of Sciences.
Terence Chi-Shen Tao is an Australian and American mathematician who is a professor of mathematics at the University of California, Los Angeles (UCLA), where he holds the James and Carol Collins Chair in the College of Letters and Sciences. His research includes topics in harmonic analysis, partial differential equations, algebraic combinatorics, arithmetic combinatorics, geometric combinatorics, probability theory, compressed sensing and analytic number theory.
Ben Joseph Green FRS is a British mathematician, specialising in combinatorics and number theory. He is the Waynflete Professor of Pure Mathematics at the University of Oxford.
In additive combinatorics, a discipline within mathematics, Freiman's theorem is a central result which indicates the approximate structure of sets whose sumset is small. It roughly states that if is small, then can be contained in a small generalized arithmetic progression.
In mathematics, a prime number p is called a Chen prime if p + 2 is either a prime or a product of two primes. The even number 2p + 2 therefore satisfies Chen's theorem.
In mathematics, an IP set is a set of natural numbers which contains all finite sums of some infinite set.
Erdős' conjecture on arithmetic progressions, often referred to as the Erdős–Turán conjecture, is a conjecture in arithmetic combinatorics. It states that if the sum of the reciprocals of the members of a set A of positive integers diverges, then A contains arbitrarily long arithmetic progressions.
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 number theory, the Green–Tao theorem, proved by Ben Green and Terence Tao in 2004, states that the sequence of prime numbers contains arbitrarily long arithmetic progressions. In other words, for every natural number k, there exist arithmetic progressions of primes with k terms. The proof is an extension of Szemerédi's theorem. The problem can be traced back to investigations of Lagrange and Waring from around 1770.
Problems involving arithmetic progressions are of interest in number theory, combinatorics, and computer science, both from theoretical and applied points of view.
Ergodic Ramsey theory is a branch of mathematics where problems motivated by additive combinatorics are proven using ergodic theory.
In mathematics, arithmetic combinatorics is a field in the intersection of number theory, combinatorics, ergodic theory and harmonic analysis.
In mathematics, in the field of additive combinatorics, a Gowers norm or uniformity norm is a class of norms on functions on a finite group or group-like object which quantify the amount of structure present, or conversely, the amount of randomness. They are used in the study of arithmetic progressions in the group. They are named after Timothy Gowers, who introduced it in his work on Szemerédi's theorem.
Tamar Debora Ziegler is an Israeli mathematician known for her work in ergodic theory, combinatorics and number theory. She holds the Henry and Manya Noskwith Chair of Mathematics at the Einstein Institute of Mathematics at the Hebrew University.
József Solymosi is a Hungarian-Canadian mathematician and a professor of mathematics at the University of British Columbia. His main research interests are arithmetic combinatorics, discrete geometry, graph theory, and combinatorial number theory.
Roth's theorem on arithmetic progressions is a result in additive combinatorics concerning the existence of arithmetic progressions in subsets of the natural numbers. It was first proven by Klaus Roth in 1953. Roth's theorem is a special case of Szemerédi's theorem for the case .
Sarah Anne Peluse is an American mathematician specializing in arithmetic combinatorics and analytic number theory, and known for her research on generalizations of Szemerédi's theorem on the existence of polynomial progressions in dense sets of integers. She is an assistant professor and LSA Collegiate Fellow in the Department of Mathematics at the University of Michigan.
