Combinatorics is a branch of mathematics concerning the study of finite or countable discrete structures.
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.
Klaus Friedrich Roth was a German-born British mathematician who won the Fields Medal for proving Roth's theorem on the Diophantine approximation of algebraic numbers. He was also a winner of the De Morgan Medal and the Sylvester Medal, and a Fellow of the Royal Society.
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.
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.
In number theory, primes in arithmetic progression are any sequence of at least three prime numbers that are consecutive terms in an arithmetic progression. An example is the sequence of primes, which is given by for .
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.
Vitaly Bergelson is a mathematical researcher and professor at Ohio State University in Columbus, Ohio. His research focuses on ergodic theory and combinatorics.
The Erdős–Turán conjecture is an old unsolved problem in additive number theory posed by Paul Erdős and Pál Turán in 1941.
In arithmetic combinatorics, the Erdős–Szemerédi theorem states that for every finite set of integers, at least one of , the set of pairwise sums or , the set of pairwise products form a significantly larger set. More precisely, the Erdős–Szemerédi theorem states that there exist positive constants c and such that for any non-empty set
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.
In mathematics, and in particular in arithmetic combinatorics, a Salem-Spencer set is a set of numbers no three of which form an arithmetic progression. Salem–Spencer sets are also called 3-AP-free sequences or progression-free sets. They have also been called non-averaging sets, but this term has also been used to denote a set of integers none of which can be obtained as the average of any subset of the other numbers. Salem-Spencer sets are named after Raphaël Salem and Donald C. Spencer, who showed in 1942 that Salem–Spencer sets can have nearly-linear size. However a later theorem of Klaus Roth shows that the size is always less than linear.
In arithmetic combinatorics, Behrend's theorem states that the subsets of the integers from 1 to in which no member of the set is a multiple of any other must have a logarithmic density that goes to zero as becomes large. The theorem is named after Felix Behrend, who published it in 1935.
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 .