On forbidden configurations in point-line incidence graphs

Abstract

The celebrated Szemer\'edi--Trotter theorem states that the maximum number of incidences between $n$ points and $n$ lines in the plane is $O(n^{4/3})$, which is asymptotically tight. Solymosi (2005) conjectured that for any set of points $P_0$ and for any set of lines $\mathcal{L}_0$ in the plane, the maximum number of incidences between $n$ points and $n$ lines in the plane whose incidence graph does not contain the incidence graph of $(P_0,\mathcal{L}_0)$ is $o(n^{4/3})$. This conjecture is mentioned in the book of Brass, Moser, and Pach (2005). Even a stronger conjecture, which states that the bound can be improved to $O(n^{4/3-\varepsilon})$ for some $\varepsilon = \varepsilon(P_0,\mathcal{L}_0)>0$, was introduced by Mirzaei and Suk (2021). We disprove both of these conjectures. We also introduce a new approach for proving the upper bound $O(n^{4/3-\varepsilon})$ on the number of incidences for configurations $(P,\mathcal{L})$ that avoid certain subconfigurations.