All Random Solved Random Open
Let $c>0$ and let $f_c(n)$ be the maximal $m$ such that every graph $G$ with $n$ vertices and at least $cn^2$ edges, where each edge is contained in at least one triangle, must contain an edge in at least $m$ different triangles. Estimate $f_c(n)$. In particular, is it true that $f_c(n)>n^{\epsilon}$ for some $\epsilon>0$? Or even $f_c(n)\gg \log n$?
A problem of Erdős and Rothschild. Alon and Trotter showed that $f_c(n)\ll_c n^{1/2}$. Szemerédi observed that his regularity lemma implies that $f_c(n)\to \infty$.

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