Determine the multiplicity covering number of the binary grid

Determine the correct lower bound, and hence the covering number, for covering the binary grid with hyperplanes so that every point is covered at least k times while one point is missed.

Background

The paper discusses the multiplicity version of the Alon–Füredi hyperplane-covering theorem, in which every point of a finite grid must be covered at least k times while one point is omitted. Ball and Serra established a general lower bound that is tight when one side of the grid is much larger than the others, but the authors note that the optimal bound is unresolved for combinatorially significant special cases, including the binary grid. The cited references [11, 16] concern this unresolved case.

References

Although this bound is tight when |S1| is much larger than |Si| for all i # 2 (see the remark after the proof of Theorem 1.1 in [6]), in some combinatorially interesting special cases like the binary grid {0,1}? it is an open problem to determine the correct bound [11, 16].

— Covering half-grids with lines and planes  (2501.11156 - Bishnoi et al., 19 Jan 2025) in Section 1, Introduction