Point-line incidence conjecture for even-cycle-free graphs

Let $k\geq 3$ be an integer. Let $P$ be a set of $n$ points in the plane and let $\mathcal{L}$ be a set of $n$ lines in the plane. Define the incidence set and incidence graph by

$$I=I(P,\mathcal{L})=\{(p,\ell)\in P\times\mathcal{L}:p\in\ell\},\qquad G=(P\cup\mathcal{L},I).$$

The incidence graph has vertex parts $P$ and $\mathcal{L}$, and is $C_{2k}$-free when it contains no cycle of length $2k$.

Point-line incidence conjecture. If $G$ is $C_{2k}$-free, then

$$|I(P,\mathcal{L})|=o\left(n^{1+1/k}\right).$$

This conjecture asks whether the geometric setting admits a bound strictly smaller than the general extremal-graph upper bound. The paper presents it as an open problem and develops lower-bound constructions for point-line arrangements with prescribed girth.