Kővári–Sós–Turán theorem
= Kővári–Sós–Turán theorem
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For each fixed positive integer $t$,
$$
\operatorname{ex}(n,K_{t,t})=O\left(n^{2-1/t}\right).
$$
The standard proof counts $t$-element subsets of <graph neighbourhood>[graph neighbourhoods] in a bipartite subgraph. If one side has degrees $d_y$, the absence of $K_{t,t}$ gives
$$
\sum_y\binom{d_y}{t}\leq(t-1)\binom nt.
$$
<convexity> then bounds the number of edges.