Erdős offered a 
prize for a proof of the proposition that "If the
sum of reciprocals of a set of integers diverges, then that set contains arbitrarily
long arithmetic progressions." This conjecture is still open (unsolved), even
for 3-term arithmetic progressions. Erdős also offered 
for an asymptotic formula for 
, the largest possible cardinality of a subset of 
that does not contain a 3-term
arithmetic progression.
Erdős-Turán Conjecture
See also
A-Sequence, B2-Sequence, Szemerédi's TheoremThis entry contributed by Kevin O'Bryant
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References
Erdős, P. and Turán, P. "On Some Sequences of Integers." J. London Math. Soc. 11, 261-264, 1936.Green, B. and Tao, T. "The Primes Contain Arbitrarily Long Arithmetic Progressions." Ann. Math. 167, 481-547, 2008. https://doi.org/10.4007/annals.2008.167.481.Referenced on Wolfram|Alpha
Erdős-Turán ConjectureCite this as:
O'Bryant, Kevin. "Erdős-Turán Conjecture." From MathWorld--A Wolfram Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/Erdos-TuranConjecture.html