Steiner general position number of the infinite grid

Determine whether \(\sgp_k(P_{\infty}\square P_{\infty})=2k\) holds for every \(k>2\).

Background

The survey presents lower bounds for Steiner general position numbers on the infinite Cartesian grid. It identifies the exact equality question beyond the already understood case as one of the open problems in Steiner general position.

References

Many open problems on the Steiner general position remain to be investigated, some of them are listed in the seminal paper paper. For instance, does the equality $\sgp_k(P_{\infty}\cp P_{\infty})=2k$ hold for $k>2$?

— The General Position Problem: A Survey  (2501.19385 - V. et al., 31 Jan 2025) in Section 4, subsection “Steiner general position”