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

## TL;DR

This paper improves the lower bounds on the constant of proportionality in point-line incidence bounds, showing constructions that achieve constants of at least 1 and approximately 1.11, surpassing previous bounds.

## Contribution

It introduces modified and new constructions that establish higher lower bounds for the constant of proportionality in incidence bounds.

## Key findings

- Elekes' construction yields a lower bound of 0.63.
- Modified Elekes' construction improves the bound to 1.
- Erdős' construction further improves the bound to approximately 1.11.

## Abstract

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, $c_{\mathrm{SzTr}} \ge 3/(2^{1/3}\pi^{2/3}) \approx 1.11$.