Improve lower bounds for fixed k

Improve the known lower bounds on $f_k(n)$ for every fixed integer $k\geq 2$, where $f_k(n)$ is the maximum size of a subset of the n×n grid having at most k collinear points on any line.

Background

Beyond the regime in which k grows with n, the paper emphasizes that little is known for constant k greater than 2. In particular, the known asymptotic lower bound for k=2 has not improved beyond approximately 1.5n, motivating the request for stronger constructions for all fixed k≥2.

References

While the lower bound on $f_2(n)$ has not been improved from the asymptotic value $1.5n$ , not much is known about $f_k(n)$ when $k>2$ is a constant.

— Settling the no-$(k+1)$-in-line problem when $k$ is not small  (2502.00176 - Kovács et al., 31 Jan 2025) in Section 6, Concluding remarks and open questions