Past exam of the mathematics course of the University of Cambridge / 2022 / ii / Paper 2 / 17F / c / Solution

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