The colouring version of this is [303], which was solved by Brown and Rödl [BrRo91]. One can take either $A$ to be all odd integers in $[1,N]$ or all integers in $[N/2,N]$ to show $f(N)\geq (1/2+o(1))N$.

Wouter van Doorn has proved, in an unpublished note, that \[f(N) \leq (9/10+o(1))N.\] Stijn Cambie has observed that \[f(N)\geq (5/8+o(1))N,\] taking $A$ to be all odd integers $\leq N/4$ and all integers in $[N/2,N]$.

Stijn Cambie has also observed that, if we allow $b=c$, then there is a solution to this equation when $\lvert A\rvert \geq (\tfrac{2}{3}+o(1))N$, since then there must exist some $n,2n\in A$.

See also [301] and [327].