Extend the exact formula to smaller k

Establish whether the equality $f_k(n)=kn$ holds when $k$ grows substantially more slowly than $C\sqrt{n\log n}$, in particular throughout ranges satisfying $k=\Omega(n^{\varepsilon})$ for every fixed $\varepsilon>0$.

Background

The main theorem proves fk(n)=knf_k(n)=kn when kk exceeds a constant multiple of nlog⁡n\sqrt{n\log n}. The authors ask whether an analogous equality remains valid for much smaller, subpolynomial-scale values of k, and explicitly conjecture that the trivial upper bound is tight in this broader range.

References

It would be very interesting to see whether a similar statement holds when $k$ is in a lower range, $k=\Omega(n{\varepsilon})$ for every $\varepsilon>0$. We pose this as an open problem.

— 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