Let $\delta_k$ be such that in any $2$-colouring of $\{1,\ldots,n\}$ there exist at least $(\delta_k+o(1))n^2$ many monochromatic $k$-term arithmetic progressions. Give reasonable bounds (or even an asymptotic formula) for $\delta_k$.

(Erdős Problem #1186 — additive combinatorics, arithmetic progressions — https://www.erdosproblems.com/1186)