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All Random Solved Random Open
OPEN - $1000
Let $r_k(N)$ be the largest possible size of a subset of $\{1,\ldots,N\}$ that does not contain any non-trivial $k$-term arithmetic progression. Prove an asymptotic formula for $r_k(N)$.
The best known bounds are due to Kelley and Meka [KeMe23] for $k=3$, Green and Tao [GrTa17] for $k=4$, and Leng, Sah, and Sawhney [LSS24] for $k\geq 5$. Erdős remarked this is 'probably unattackable at present'. Given that Erdős elsewhere offered \$5000 just for (essentially) showing that $r_k(N)=o_k(N/\log N)$, the value of this prize seems odd.

See also [3] and [139].

Additional thanks to: Zachary Hunter