On forbidden configurations in point-line incidence graphs

The celebrated Szemeredi-Trotter theorem states that the maximum number of incidences between $n$ points and $n$ lines in the plane is $\mathcal{O}(n^{4/3})$, which is asymptotically tight.

Solymosi conjectured that this bound drops to $o(n^{4/3})$ if we exclude subconfigurations isomorphic to any fixed point-line configuration. We describe a construction disproving this conjecture. On the other hand, we prove new upper bounds on the number of incidences for configurations that avoid certain subconfigurations. Joint work with Martin Balko.