Erd\H{o}s and Rothschild asked to estimate the maximum number, denoted by $h(n,c)$, such that every $n$-vertex graph with at least $cn^2$ edges, each of which is contained in at least one triangle, must contain an edge that is in at least $h(n,c)$ triangles. In particular, Erd\H{o}s asked in 1987 to determine whether for every $c>0$ there is $\epsilon>0$ such that $h(n,c)>n^{\epsilon}$ for all sufficiently large $n$. We prove that $h(n,c)=n^{O(1/\log \log n)}$ for every fixed $c<1/4$. This gives a negative answer to the question of Erd\H{o}s, and is best possible in terms of the range for $c$, as it is known that every $n$-vertex graph with more than $n^2/4$ edges contains an edge that is in at least $n/6$ triangles.

Joint work with Jacob Fox.