A small inquiry about something that has been troubling me for the whole afternoon without luck: is there any known result about say simple graphs $G(V,E)$ with some property $\mathcal{P}$ such that the number of triangles $t(G)$ is bounded above by $O(|V|^{\frac{3}{2}})$?


Sorry if it is not MO appropriate :).



An upper bound for number of triangles in a graph
The property is that the graph be sparse, since it is easy to show that the number of triangles is $O(|E|^{3/2}),$ so as long as $E = O(V),$ your result holds. For the (simple) proof and sharp extensions see


Rivin, Igor(1-TMPL)
Counting cycles and finite dimensional Lp norms. (English summary) 
Adv. in Appl. Math. 29 (2002), no. 4, 647–662. 
05C38 (90C35) 


