Erdős–Turán Conjecture (other)
Erdős–Turán conjecture may refer to: * Szemerédi's theorem * Erdős conjecture on arithmetic progressions * Erdős–Turán conjecture on additive bases The Erdős–Turán conjecture is an old unsolved problem in additive number theory (not to be confused with Erdős conjecture on arithmetic progressions) posed by Paul Erdős and Pál Turán in 1941. It concerns additive bases, subsets of natu ...

Szemerédi's Theorem
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. Statement A subset ''A'' of the natural numbers is said to have positive upper density if :\limsup_\frac > 0. Szemerédi's theorem asserts that a subset of the natural numbers with positive upper density contains an arithmetic progression of length ''k'' for all positive integers ''k''. An often-used equivalent finitary version of the theorem states that for every positive integer ''k'' and real number \delta \in (0, 1], there exists a positive integer :N = N(k,\delta) such that every subset of of size at least \delta N contains an arithmetic progression of length ''k''. Another formulation uses the function ''r''''k''(''N''), the size of ...
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Erdős Conjecture On Arithmetic Progressions
Erdős' conjecture on arithmetic progressions, often referred to as the Erdős–Turán conjecture, is a conjecture in arithmetic combinatorics (not to be confused with the Erdős–Turán conjecture on additive bases). 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. Formally, the conjecture states that if ''A'' is a large set in the sense that \sum_ \frac \ =\ \infty, then ''A'' contains arithmetic progressions of any given length, meaning that for every positive integer ''k'' there are an integer ''a'' and a non-zero integer ''c'' such that \\subset A. History In 1936, Erdős and Turán made the weaker conjecture that any set of integers with positive natural density contains infinitely many three-term arithmetic progressions.. This was proven by Klaus Roth in 1952, and generalized to arbitrarily long arithmetic progressions by Szemerédi in 1975 in wha ...
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