Erd\H{o}s remarked this is 'probably unattackable at present'. In \cite{Er97c} Erd\H{o}s offered \$1000, but given that he elsewhere offered \$5000 just for (essentially) showing that $r_k(N)=o_k(N/\log N)$, that value seems odd. In \cite{Er81} he offers \$10000, stating it is 'probably enormously difficult'.<br><br>The best known upper bounds for $r_k(N)$ are due to Kelley and Meka \cite{KeMe23} for $k=3$, Green and Tao \cite{GrTa17} for $k=4$, and Leng, Sah, and Sawhney \cite{LSS24} for $k\geq 5$. An asymptotic formula is still far out of reach, even for $k=3$.
  
References
[Er81] Erd\H{o}s, P., <i>On the combinatorial problems which I would most like to see solved</i>. Combinatorica (1981), 25-42.
[Er97c] Erd\H{o}s, Paul, <i>Some of my favorite problems and results</i>. The mathematics of Paul Erd\H{o}s, I (1997), 47-67.
[GrTa17] Green, Ben and Tao, Terence, <i>New bounds for Szemer\'{e}di's theorem, III: a polylogarithmic bound for $r_4(N)$</i>. Mathematika (2017), 944-1040.
[KeMe23] Kelley, Z. and Meka, R., <i>Strong Bounds for 3-Progressions</i>. arXiv:2302.05537 (2023).
[LSS24] Leng, J., Sah, A. and Sawhney, M., <i>Improved bounds for Szemer\'{e}di's theorem</i>. arXiv:2402.17995 (2024).