= Incidence bound from two-point multiplicity
{title2=$I\le m+\sqrt{\lambda mn(n-1)}$}
If $n$ points and $m$ curves have the property that each pair of distinct points lies on at most $\lambda$ curves, then <double counting> gives $\sum_\gamma k_\gamma(k_\gamma-1)\le\lambda n(n-1)$, where $k_\gamma=|P\cap\gamma|$. The <Cauchy-Schwarz inequality> implies $I^2\le mI+\lambda mn(n-1)$ and hence $I\le m+\sqrt{\lambda mn(n-1)}$. The statement needs only this combinatorial multiplicity condition, not algebraicity.
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