Conlon–Fox–Sudakov conjecture on books versus triangles

Let $G$ be a graph on $n$ vertices, let $b$ be a real parameter satisfying

$$\frac{n}{6}\le b<\frac{n}{4},$$

and suppose that $G$ has at least $\lfloor n^2/4\rfloor$ edges. The book number $b(G)$ is the maximum number of triangles sharing a common edge; the balanced complete bipartite graph is the complete bipartite graph with parts as equal in size as possible. Let $S_{b,n}$ denote the specified blow-up of the $3$-prism.

Conlon–Fox–Sudakov conjecture. If $b(G)\le b$ and $G$ is not the balanced complete bipartite graph, then $G$ has at least

$$b^2(n-4b)$$

triangles, with equality if and only if $G$ is $S_{b,n}$.

The conjecture gives an exact description of the tradeoff between the local parameter $b(G)$ and the total number of triangles in the range $n/6\le b<n/4$, complementing asymptotic results for larger book numbers. The lower endpoint is motivated by the known bound $b(G)\ge n/6$ for graphs with at least $\lfloor n^2/4\rfloor$ edges.