Rhombus-free edge bound

A graph-theoretic rhombus is two triangles sharing an edge. Every rhombus-free graph on $n\ge 4$ vertices has at mostedges. If the graph is triangle-free this is Turán's theorem. Otherwise remove a triangle: every remaining vertex has at most one neighbor in it, and induction bounds the edge count by
$\lfloor (n-3{)}^2/4\rfloor +n\le \lfloor n^2/4\rfloor$.

The triangular prism has six vertices and nine edges. Each edge lies in at most one triangle, so it has no rhombus; its two triangles show that it is not $K_{3,3}=T_2(6)$.