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)$\mathrm{.}Let$c_{$\mathrm{S}\mathrm{z}\mathrm{T}\mathrm{r}$}$ denote the lower bound on the constant of proportionality of the $n^{2\mathrm{/}3}{l}^{2\mathrm{/}3}$ term. The known lower bound, due to Elekes, is $c_{\mathrm{S}\mathrm{z}\mathrm{T}\mathrm{r}}\ge 2^{-2\mathrm{/}3}=0.63$. With a slight modification of Elekes’ construction, we show that it can give a better lower bound of $c_{\mathrm{S}\mathrm{z}\mathrm{T}\mathrm{r}}\ge 1$, i.e., $I(n,l)\ge n^{2\mathrm{/}3}{l}^{2\mathrm{/}3}$. Furthermore, we analyze a different construction given by Erd{\H o}s, and show its constant of proportionality to be even better, $c_{$\mathrm{S}\mathrm{z}\mathrm{T}\mathrm{r}$} \ge 3/(2^{1/3}\pi^{2/3}) \approx 1.11$.