Lower bounds for incidences with hypersurfaces (Q2826220)

The author gives a technique for deriving lower bounds for incidences with various families of geometric objects in \(\mathbb R^d\) where \(d \geq 4.\) These families include hyperplanes, hypersurfaces, paraboloids and sets that are not algebraic. Specifically, the author proves the following. Let \(\varepsilon, \delta >0\) and let \(m\) and \(d\) be positive integers with \(d \geq 4\). Then there exists a set \(P\) of \(m\) points and a set \(\Pi\) of \(n = \Theta(m^{(3-3\varepsilon)/(d+1)} ) \) hyperplanes (hyperspheres or any linear-closed family of graphs), both in \(\mathbb R^d\), such that the incidence graph of \(P \times \Pi\) contains no copy of \(K_{2,d/\varepsilon} \) and with \(I(P,\Pi) = \Omega(m^\delta n^{ (d+2 - \delta(d+1))/3 - \varepsilon}).\)