ERDőS CONJECTURE ON ARITHMETIC PROGRESSIONS

'Erdős' conjecture on arithmetic progressions', often ''incorrectly'' referred to as the Erdős–Turán conjecture, is an unproven proposition in additive combinatorics. The conjecture 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, if
:
then ''A'' contains arithmetic progressions of any given length.
If true, the theorem would generalize Szemerédi's theorem.
The Green-Tao theorem on arithmetic progressions in the primes is a special case of this conjecture.

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References

References

★ P. Erdős: Résultats et problèmes en théorie de nombres, ''Séminaire Delange-Pisot-Poitou (14e année: 1972/1973), Théorie des nombres'', Fasc 2., Exp. No. 24, pp. 7,

★ P. Erdős: Problems in number theory and combinatorics, Proc. Sixth Manitoba Conf. on Num. Math., ''Congress Numer.'' 'XVIII'(1977), 35-58.

★ P. Erdős: On the combinatorial problems which I would most like to see solved, ''Combinatorica'', '1'(1981), 28.