A problem of Erdős and Rothschild. Alon and Trotter showed that, provided $c<1/4$, $f_c(n)\ll_c n^{1/2}$. Szemerédi observed that his regularity lemma implies that $f_c(n)\to \infty$.

Edwards (unpublished) and Khadziivanov and Nikiforov [KhNi79] proved independently that $f_c(n) \geq n/6$ when $c>1/4$ (see [905]).

Fox and Loh [FoLo12] proved that \[f_c(n) \leq n^{O(1/\log\log n)}\] for all $c<1/4$, disproving the first conjecture of Erdős.

The best known lower bounds for $f_c(n)$ are those from Szemerédi's regularity lemma, and as such remain very poor.

See also [600] and the entry in the graphs problem collection.