The Maximum Number of Triangles in a Graph of Given Maximum Degree

We prove that any graph on $n$ vertices with max degree $d$ has at most $q{d+1 \choose 3}+{r \choose 3}$ triangles, where $n = q(d+1)+r$, $0 \le r \le d$. This resolves a conjecture of Gan-Loh-Sudakov.

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