Characterize the asymptotic value of the No-Three-In-Line problem

Characterize the asymptotic maximum size of a subset of the n×n square lattice containing no three collinear points, distinguishing whether the leading-order value is approximately 1.5n, 2n, or an intermediate quantity.

Background

The paper notes that existing constructions provide approximately 1.5n points, while 2n is the general upper bound. It further explains that the asymptotic behavior is unsettled, with different possibilities advocated in the literature, so even the appropriate conjectural leading constant is not established.

References

Interestingly, even what to conjecture on the asymptotical value of the answer for the No-Three-In-Line problem is far from clear, with prominent experts believing it could be roughly $1.5n$ via the previous construction (see Green ), or it could be $2n$ thus the upper bound can be attained asymptotically in general , or somewhere in between .

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