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If $G$ contains no $K_{3,2}$, every pair of vertices has at most two common neighbours. Double-counting a vertex together with an unordered pair of its neighbours givesConsequentlyBy Cauchy--Schwarz,This quadratic inequality implies $e<c{n}^{3/2}$ for an absolute constant $c$; for example $c=2$ works for every $n\ge 1$. Thus