The Constant of Proportionality in Lower Bound Constructions of Point-Line Incidences

Let $I(n,l)$ denote the maximum possible number of incidences between $n$ points and $l$ lines. It is well known that $I(n,l) = \Theta(n^{2/3}l^{2/3} + n + l)$. Let $c_{\mathrm{SzTr}}$ denote the lower bound on the constant of proportionality of the $n^{2/3}l^{2/3}$ term. The known lower bound, due to Elekes, is $c_{\mathrm{SzTr}} \ge 2^{-2/3} = 0.63$. With a slight modification of Elekes' construction, we show that it can give a better lower bound of $c_{\mathrm{SzTr}} \ge 1$, i.e., $I(n,l) \ge n^{2/3}l^{2/3}$. Furthermore, we analyze a different construction given by Erd{\H o}s, and show its constant of proportionality to be even better, $c_{\mathrm{SzTr}} \ge 3/(2^{1/3}\pi^{2/3}) \approx 1.11$.